User:Tomruen/polychoric groups

Conway

TEXT FROM (And extended tables by me for Coxeter's real notations, and group orders calculated)

• Conway and Smith: On Quaternions and Octonions, Chapter 4, section 4.4 Coxeter's Notations for the Polyhedral Groups

Coxeter's notations (adapted from Schlafli) for regular polytopes and associated groups are widely used. Conway extended his system slightly to obtain a complete set of notations for the "polyhedral groups".

Coxeter uses [p,q,...,r,s] for the symmetry group of the n-dimensional polytope {p,q,...,r,s} - this is generated by reflections R1, ... Rn, corresponding to the nodes of the n-node Coxeter diagram: ....

This has an obvious subgroup [p,q,...,r,s]⁺ of index 2, consisting of the even length words in R1...Rn, i.e. those which "mention R1..Rn evenly."

When just one of the numbers p,q,...r,s is even say that between Rk and Rk+1, there exists two further subgroups, namely... [p⁺,q,...,r,s] and [p,q,...,r,s⁺] consisting of words that mention respectively R1...Rk, and Rk+1..Rn evenly, whose intersection is the index 4 subgroup [⁺p,q,...,r,s⁺] of words that mention both of these sets evenly. We've slightly modified Coxeter's notation - he writes [p⁺...] and uses only some special cases.

To obtain elegant names for all of the polyhedral groups in dimension 4, we supplement Coxeter's notation by writing G^o for the "opposite" group to G, obtained by replacing the element g of G by +g, or -g accordingly as det = +1 or -1. Finally an initial "2." indicates doubling the group by adjoining negatives, while a final ":2" or "2" indicates doubling in some other way, either split or non-split.

± implies central inversion (double rotation), trailing .2 implies reflective symmetry

Chiral I Group Order DuV Coxeter ±[I×O] 2880 #29 2/5[5,3,3]+ ±[I×T] 1440 #24 1/5[5,3,3]+ ±[I×D2n] 240n #19 2n/120[5,3,3]+ ±[I×Cn] 120n #9 n/120[5,3,3]+ ±[O×T] 576 #23 [[3+,4,3+]] ±[O×D2n] 96n #15 ±1/2[O×D2n] 48n #16 ±1/2[O×D4n] 96n #17 ±1/6[O×D6n] 48n #18 ±[O×Cn] 48n #7 ±1/2[O×C2n] 48n #8 ±[T×D2n] 48n #14 ±[T×Cn] 24n #5 ±1/3[T×C3n] 24n #6

Chiral II Achiral Group Order DuV Coxeter (Conway) Coxeter (Real) Group Order DuV Coxeter (Conway) Coxeter (Real) ±[I×I] 7200 #30 [3,3,5]+ ±[I×I].2 14400 #50 [3,3,5] ±1/60[I×I] 120 #31 2.[3,5]+ [2,3,5]+ ±1/60[I×I].2 240 #49 2.[3,5] [2,3,5] +1/60[I×I] 60 #31' [3,5]+ +1/60[I×I].23 120 #49" [3,5] +1/60[I×I].21 120 #49' [3,5]o [2,(3,5)+] ±1/60[I×I] 120 #32 2.[3,3,3]+ [[3,3,3]]+ ±1/60[I×I].2 240 #51 2.[3,3,3] [[3,3,3]] +1/60[I×I] 60 #32' [3,3,3]+ +1/60[I×I].23 120 #51" [3,3,3]o [[3,3,3]+] +1/60[I×I].21 120 #51' [3,3,3] ±[O×O] 1152 #25 [3,4,3]+:2 [[3,4,3]]+ ±[O×O].2 2304 #48 [3,4,3]:2 [[3,4,3]] ±1/2[O×O] 576 #28 [3,4,3]+ ±1/2[O×O].23 1152 #45 [3,4,3] ±1/2[O×O].2 #46 [3,4,3]+.2 [[3,4,3]+] ±1/6[O×O] 192 #27 [3,3,4]+ ±1/6[O×O].2 384 #47 [3,3,4] ±1/24[O×O] 48 #26 2.[3,4]+ [2,3,4]+ ±1/24[O×O].2 96 #44 2.[3,4] [2,3,4] +1/24[O×O] 24 #26' [3,4]+ +1/24[O×O].23 48 #44" [3,4] +1/24[O×O].21 48 #44' [3,4]o [2,(3,4)+] +1/24[O×O] 24 #26" [2,3,3]+ +1/24[O×O].23 48 #40" [2,3,3] +1/24[O×O].21 48 #40' [2,3,3]o [(2,3)+,4] ±[T×T] 288 #20 [+3,4,3+] [3+,4,3+] ±[T×T].2 576 #43 [3,4,3+] ±1/3[T×T] 96 #22 [+3,3,4+] [(3,3)+,4,1+] =[31,1,1]+ ±1/3[T×T].2 192 #41 [+3,3,4] [(3,3)+,4] ≈ ±1/3[T×T] ±1/3[T×T].2 192 #42 [3,3,4+] [3,3,4,1+] ±1/12[T×T] 24 #21 2.[3,3]+ [3+,4,2+] ±1/12[T×T].2 48 #39 2.[3+,4] [3+,4,2] ≈ ±1/12[T×T] ±1/12[T×T].2 48 #39' 2.[3,3] [3,4,2+] +1/12[T×T] 12 #21' [3,3]+ +1/12[T×T].23 24 [3+,4] +1/12[T×T].21 24 [3+,4]o [(3,4)+,2+] ≈ +1/12[T×T] +1/12[T×T].23 24 #39' [3,3]o [(3,3)+,2] +1/12[T×T].21 24 #39" [3,3]

Duoprismatic

Duoprismatic

Chiral I Group Order DuV Coxeter ±1/2[D2m×D4n] 8mn #13 ±[D2m×Cn] 4mn ±1/2[D2m×C2n] 4mn +1/2[D2m×C2n] 2mn ±1/2[D4m×C2n] 8mn

Chiral II Achiral Group Order DuV Coxeter Group Order DuV Coxeter ±[D2m×D2n] 8mn [2m+,2,2n+] ±[D2n×D2n].2 16n2 ±1/2[D4m×D4n] 16mn ±1/2[D4n×D4n].2 32n2 ±1/2[D4n×D4n].2 ±1/4[D4m×D4n] 8mn ±1/4[D4n×D4n].2 16n2 ±1/2[D2m×D2n] 4mn #11 [4m+,2+,4n+] ±1/2[D2n×D2n].2 8n2 ±1/2[D2n×D2n].2 ±[Cm×Cn] 2mn #1 [2m+,2+,2n+] ±[Cn×Cn].2 4n2 m,n odd +1/4[D4m×D4n] 4mn +1/4[D4n×D4n].23 8n2 +1/4[D4nxD4n].21 +1/2[D2m×D2n] 2mn #11' [4m+,2+,4n+]+ +1/2[D2n×D2n].2 4n2 +1/2[D2n×D2n].2 +[Cm×Cn] mn #1' [2m+,2+,2n+]+ +[Cn×Cn].2 2n2

Chiral II			Achiral
±1/2f[D2f×D2f(s)]	4f	#11	±1/2f[D2f×D2f(s)].2(α,β)	8f
			±1/2f[D2f×D2f(s)].2
+1/2f[D2f×D2f(s)]	2f	#11'	+1/2f[D2f×D2f(s)].2(α,β)	4f
			+1/2f[D2f×D2f(s)].2
±1/f[Cf×Cf(s)]	2f	#1	±1/f[Cf×Cf(s)].2(γ)	4f
+1/f[Cf×Cf(s)]	f	#1'	+1/f[Cf×Cf(s)].2(γ)	2f

m,n odd

Chiral II			Achiral
±1/2f[D2mf×D2nf(s)]	4mnf	#11	±1/2f[D2nf×D2nf(s)].2(α,β)	8n2f
			±1/2f[D2nf×D2nf(s)].2
+1/2f[D2mf×D2nf(s)]	2mnf	#11'	+1/2f[D2nf×D2nf(s)].2(α,β)	4n2f
			+1/2f[D2nf×D2nf(s)].2
±1/f[Cmf×Cnf(s)]	2mnf	#1	±1/f[Cnf×Cnf(s)].2(γ)	4n2f
+1/f[Cmf×Cnf(s)]	mnf	#1'	+1/f[Cnf×Cnf(s)].2(γ)	2n2f

Coxeter

From:

• H.S.M. Coxeter: Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]

Symbol	Order.index	Du Val's symbol	Conway	Structure
[3,3,2]+	24.10	(T/C2;T/C2)	+1/24[O×O]	C2×A4
[(3,3)+,2]	24.10	(T/C1;T/C1)c*	±1/12[T×T].21	C2×A4
[[3,2,3]+]	36.14	(D3/C3;D3/C3)*		(4,4|2,3)
[4,3,2]+	48.36	(O/C2;O/C2)	±1/24[O×O]	C2×S4
[3,3,2]	48.36	(O/C1;O/C1)*	+1/24[O×O].23	C2×S4
[(4,3)+,2]	48.36	(O/C1;O/C1)-*	+1/24[O×O].21	C2×S4
[3+,4,2]	48.22	(T/C2;T/C2)c*	±1/12[T×T].2	D2×A4
[3,3,3]+	60.13	(I†/C1;I/C1)†	+1/60[I×I]	A5
[1+,4,3,3]+	96.1	(T/V;T/V)	±1/3[T×T]	?
[4,3,2]	96.5	(O/C2;O/C2)*	±1/24[O×O].2	D2×S4
[3,3,3]	120.1	(I†/C1;I/C1)†*	+1/60[I×I].23	S5=(4,6|2,3)
[[3,3,3]+]	120.1	(I†/C1;I/C1)-†*	+1/60[I×I].21	S5
[[3,3,3]]+	120.2	(I†/C2;I/C2)†	±1/60[I×I]	C2×A5
[5,3,2]+	120 (non-c)	(I/C2;I/C2)	±1/60[I×I]	C2×A5
[(5,3)+,2]	120 (non-c)	(I/C1;I/C1)*	+1/60[I×I].21	C2×A5
[4,3,3]+	192.3	(O/V;O/V)	±1/6[O×O]	?
[(3,3)+,4]	192.1	(T/V;T/V)*	±1/3[T×T].2	?
[3,3,4,1+]	192.2	(T/V;T/V)-*	±1/3[T×T].2	?
[[3,3,3]]	240.1	(I†/C2;I/C2)†*	±1/60[I×I].2	C2×S5
[5,3,2]	240 (non-c.)	(I/C2;I/C2)*	±1/60[I×I].2	D2×A5
[3+,4,3+]	288.1	(T/T;T/T)	±[T×T]	?
[4,3,3]	384.1	(O/V;O/V)*	±1/6[O×O].2	C2^S4
[3,4,3]+	576.2	(O/T;O/T)	±1/2[O×O]	?
[3+,4,3]	576.1	(T/T;T/T)*	±[T×T].2	?
[[3+,4,3+]]	576 (non-c)	(T/T;O/O)	±[O×T]	?
[3,4,3]	1152.1	(O/T;O/T)*	[O×O].23	?
[[3,4,3]+]	1152 (non-c)	(O/T;O/T)-*	±1/2[O×O].2	(4,8|2,3)
[[3,4,3]]+	1152 (non-c)	(O/O;O/O)	±[O×O]	?
[[3,4,3]]	2304 (non-c)	(O/O;O/O)*	±[O×O].2	?
[5,3,3]+	7200 (non-c)	(I/I;I/I)	±[I×I]	?
[5,3,3]	14400 (non-c)	(I/I;I/I)*	±[I×I].2	?

(non-c) = noncrystalographic (4-space symmetry)

By reflective groups

Finite groups
[ ]: Symbol Order [1]+ 1.1 [1] = [ ] 2.1 [2]: Symbol Order [1+,2]+ 1.1 [2]+ 2.1 [2] 4.1 [2,2]: Symbol Order [2+,2+]+ = [(2+,2+,2+)] 1.1 [2+,2+] 2.1 [2,2]+ 4.1 [2+,2] 4.1 [2,2] 8.1 [2,2,2]: Symbol Order [(2+,2+,2+,2+)] = [2+,2+,2+]+ 1.1 [2+,2+,2+] 2.1 [2+,2,2+] 4.1 [(2,2)+,2+] 4 [[2+,2+,2+]] 4 [2,2,2]+ 8 [2+,2,2] 8.1 [(2,2)+,2] 8 [[2+,2,2+]] 8.1 [2,2,2] 16.1 [[2,2,2]]+ 16 [[2,2+,2]] 16 [[2,2,2]] 32	[p]: Symbol Order [p]+ p [p] 2p [p,2]: Symbol Order [p,2]+ 2p [p,2] 4p [2p,2+]: Symbol Order [2p,2+] 4p [2p+,2+] 2p [p,2,2]: Symbol Order [p+,2,2+] 2p [(p,2)+,2+] 2p [p,2,2]+ 4p [p,2,2+] 4p [p+,2,2] 4p [(p,2)+,2] 4p [p,2,2] 8p [2p,2+,2]: Symbol Order [2p+,2+,2+]+ p [2p+,2+,2+] 2p [2p+,2+,2] 4p [2p+,(2,2)+] 4p [2p,(2,2)+] 8p [2p,2+,2] 8p	[p,2,q]: Symbol Order [p+,2,q+] pq [p,2,q]+ 2pq [p+,2,q] 2pq [p,2,q] 4pq [(p,2)+,2q]: Symbol Order [(p,2)+,2q+] 2pq [(p,2)+,2q] 4pq [2p,2,2q]: Symbol Order [2p+,2+,2q+]+= [(2p+,2+,2q+,2+)] pq [2p+,2+,2q+] 2pq [2p,2+,2q+] 4pq [((2p,2)+,(2q,2)+)] 4pq [2p,2+,2q] 8pq [[p,2,p]]: Symbol Order [[p+,2,p+]] 2p2 [[p,2,p]]+ 4p2 [[p,2,p]+] 4p2 [[p,2,p]] 8p2 [[2p,2,2p]]: Symbol Order [[(2p+,2+,2p+,2+)]] 2p2 [[2p+,2+,2p+]] 4p2 [[((2p,2)+,(2p,2)+)]] 8p2 [[2p,2+,2p]] 16p2	[3,3,2]: Symbol Order [(3,3)Δ,2,1+] ≅ [2,2]+ 4 [(3,3)Δ,2] ≅ [2,(2,2)+] 8 [(3,3)⅄,2,1+] ≅ [4,2+] 8 [(3,3)+,2,1+] = [3,3]+ 12.5 [(3,3)⅄,2] ≅ [2,4,2+] 16 [3,3,2,1+] = [3,3] 24 [(3,3)+,2] 24.10 [3,3,2]+ 24.10 [3,3,2] 48.36 [4,3,2]: Symbol Order [1+,4,3+,2,1+] = [3,3]+ 12 [3+,4,2+] 24 [(3,4)+,2+] 24 [1+,4,3+,2] = [(3,3)+,2] 24.10 [3+,4,2,1+] = [3+,4] 24.10 [(4,3)+,2,1+] = [4,3]+ 24.15 [1+,4,3,2,1+] = [3,3] 24 [1+,4,(3,2)+] = [3,3,2]+ 24 [3,4,2+] 48 [4,3+,2] 48.22 [4,(3,2)+] 48 [(4,3)+,2] 48.36 [1+,4,3,2] = [3,3,2] 48.36 [4,3,2,1+] = [4,3] 48.36 [4,3,2]+ 48.36 [4,3,2] 96.5 [5,3,2]: Symbol Order [(5,3)+,2,1+] = [5,3]+ 60.13 [5,3,2,1+] = [5,3] 120.2 [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 (nc)	[31,1,1]: Symbol Order [31,1,1]Δ ≅[[4,2+,4]]+ 32 [31,1,1]⅄ 64 [31,1,1]+ 96.1 [31,1,1] 192.2 <[3,31,1]> = [4,3,3] 384.1 [3[31,1,1]] = [3,4,3] 1152.1 [3,3,3]: Symbol Order [3,3,3]+ 60.13 [3,3,3] 120.1 [[3,3,3]]+ 120.2 [[3,3,3]+] 120.1 [[3,3,3]] 240.1 [4,3,3]: Symbol Order [1+,4,(3,3)Δ] = [31,1,1]Δ ≅[[4,2+,4]]+ 32 [4,(3,3)Δ] = [2+,4[2,2,2]+] ≅[[4,2+,4]] 64 [1+,4,(3,3)⅄] = [31,1,1]⅄ 64 [1+,4,(3,3)+] = [31,1,1]+ 96.1 [4,(3,3)⅄] ≅ [[4,2,4]] 128 [1+,4,3,3] = [31,1,1] 192.2 [4,(3,3)+] 192.1 [4,3,3]+ 192.3 [4,3,3] 384.1 [3,4,3]: Symbol Order [3+,4,3+] 288.1 [3,4,3⅄] = [4,3,3] 384.1 [3,4,3]+ 576.2 [3+,4,3] 576.1 [[3+,4,3+]] 576 (nc) [3,4,3] 1152.1 [[3,4,3]]+ 1152 (nc) [[3,4,3]+] 1152 (nc) [[3,4,3]] 2304 (nc) [5,3,3]: Symbol Order [5,3,3]+ 7200 (nc) [5,3,3] 14400 (nc)