C₂×#130848 #72310730

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

Aliases

direct product

C₂×#130848

Subgroups

	subgroup	G/H
center, socle	C₂²	#8376
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	C₂×#61267	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

3943 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:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231596, C₂×#72231719, C₂×#72231701
maximal normal subgroup of	C₂×#72231596, C₂×#72231719, C₂×#72231701

GAP

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

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

C2×#130848 #72310730