C₂×#140192 #72320107

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#140192

Subgroups

	subgroup	G/H
center	C₂×C₄	#1936
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	#9556
prime radical	C₂²×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1233 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224949, C₂×#72224958, C₂×#72224959
maximal normal subgroup of	C₂×#72224949, C₂×#72224958, C₂×#72224959

GAP

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

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

C2×#140192 #72320107