C₂×#130845 #72310759

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#130845

Subgroups

	subgroup	G/H
center, socle	C₂²	#8376
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	C₂×#61288	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(4,10,15,27)(5,16)(6,7)(9,20)(11,12,25,23)(13,24,30,28)(14,31,29,26)(18,32)(19,33)(21,22);(4,11,15,25)(10,23,27,12)(13,31,30,26)(14,28,29,24);(3,4,7,13,20,29,21,23,8,15,22,30,9,14,6,12)(5,10,18,25,33,31,34,28,19,27,32,11,16,26,17,24);(1,2)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231719, C₂×#72231593, C₂×#72231708
maximal normal subgroup of	C₂×#72231719, C₂×#72231593, C₂×#72231708

GAP

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

gap> G := Group((4,10,15,27)(5,16)(6,7)(9,20)(11,12,25,23)(13,24,30,28)(14,31,29,26)(18,32)(19,33)(21,22), (4,11,15,25)(10,23,27,12)(13,31,30,26)(14,28,29,24), (3,4,7,13,20,29,21,23,8,15,22,30,9,14,6,12)(5,10,18,25,33,31,34,28,19,27,32,11,16,26,17,24), (1,2));

C2×#130845 #72310759