C₂×#135792 #72226183

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

Aliases

direct product

C₂×#135792

Subgroups

	subgroup	G/H
center, socle	C₂³	#1844
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂⁴	#498
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#135792, C₂×#72213975, C₂×#72214006, …
maximal normal subgroup of	C₂×#72213975, C₂×#72214006, C₂×#72214020
maximal quotient of	D₃×#135792

GAP

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

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

C2×#135792 #72226183