C₂×#72217187 #72322278

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#72217187

Subgroups

	subgroup	G/H
center	C₂×C₈	#1455
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₂×#346	C₂⁴
socle	C₂²	#128278
prime radical	C₂⁵	#347
LCS	G ▷ C₂²×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(10,29)(15,37)(30,32)(31,59)(38,40)(39,63)(44,64)(50,60);(1,2);(3,4)(5,15,16,38,18,37,20,40)(6,21,22,51)(7,25)(8,14)(9,11)(10,19,30,43,29,48,32,49)(12,24,33,41)(13,27)(17,44,45,63,46,64,42,39)(23,54,53,52)(26,57)(28,34)(31,61,50,66,59,65,60,47)(35,62,56,55)(36,58);(3,5,6,17,7,19,23,47,8,20,24,45,26,49,55,61,9,18,22,46,27,48,53,66,28,16,41,42,58,43,62,65)(4,10,12,31,13,15,35,39,14,32,21,50,36,40,52,44,11,29,33,59,25,37,56,63,34,30,51,60,57,38,54,64)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225066, C₂×#72217249, C₂×#72225085
maximal normal subgroup of	C₂×#72225066, C₂×#72217249, C₂×#72225085

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((10,29)(15,37)(30,32)(31,59)(38,40)(39,63)(44,64)(50,60), (1,2), (3,4)(5,15,16,38,18,37,20,40)(6,21,22,51)(7,25)(8,14)(9,11)(10,19,30,43,29,48,32,49)(12,24,33,41)(13,27)(17,44,45,63,46,64,42,39)(23,54,53,52)(26,57)(28,34)(31,61,50,66,59,65,60,47)(35,62,56,55)(36,58), (3,5,6,17,7,19,23,47,8,20,24,45,26,49,55,61,9,18,22,46,27,48,53,66,28,16,41,42,58,43,62,65)(4,10,12,31,13,15,35,39,14,32,21,50,36,40,52,44,11,29,33,59,25,37,56,63,34,30,51,60,57,38,54,64));

C2×#72217187 #72322278