C₂×#122220 #72225147

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#122220

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂³	C₂×#322
prime radical	C₂⁴	#322
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1015 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214029, #72225261, #72225292, …
maximal normal subgroup of	C₂×#72214029, #72225261, #72225292, …
maximal quotient of	D₃×#122220

GAP

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

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

C2×#122220 #72225147