C₂×#150131 #72321226

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#150131

Subgroups

	subgroup	G/H
center, socle	C₂²	#11654
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	#346	C₂⁴
prime radical	C₂×#2218	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1008 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225113, C₂×#72225114, C₂×#72225043
maximal normal subgroup of	C₂×#72225113, C₂×#72225114, C₂×#72225043

GAP

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

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

C2×#150131 #72321226