C₂×#72224659 #72245339

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

Aliases

direct product

C₂×#72224659

Subgroups

	subgroup	G/H
center, socle	C₂²	#93757
commutator	C₃₂	C₂²×C₁₆
Frattini, cosocle	#7330	C₂³
prime radical	C₂²×D₃₂	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

488 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);(3,4,5,14,7,16,24,51,9,18,25,47,8,38,39,56,35,50,63,49,41,54,33,64,40,19,34,53,65,17,58,15,12,21,28,48,37,57,32,55,23,61,62,60,42,66,22,45,11,20,27,13,36,44,6,29,10,43,26,46,30,59,31,52);(4,13,21,49)(5,22)(6,26)(7,30)(8,36)(9,10)(11,35)(14,50,48,20)(15,54,52,44)(16,56,57,45)(17,29,59,64)(18,60,61,47)(19,53,43,46)(23,40)(24,62)(25,32)(27,31)(28,39)(33,34)(37,65)(38,51,66,55)(41,42)(58,63)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224662, D₃×#72224659, #72286570, …
maximal normal subgroup of	C₂×#72224662, #72286570, #72287062, …
maximal quotient of	D₃×#72224659, #72287264

GAP

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

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

C2×#72224659 #72245339