C₂×#72224654 #72245335

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#72224654

Subgroups

	subgroup	G/H
center, socle	C₂²	#93757
commutator	C₃₂	C₂²×C₁₆
Frattini, cosocle	#7330	C₂³
prime radical	C₂×#2262	C₈
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ #7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 66 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224662, D₃×#72224654, #72287138, …
maximal normal subgroup of	C₂×#72224662, #72287138, #72287177, …
maximal quotient of	D₃×#72224654, #72287177

GAP

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

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

C2×#72224654 #72245335