C₂×#72278057 #72297504

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂×#72278057

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂²×#15947	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8848 conjugacy classes of subgroups; only the 12240 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

(7,8)(9,10)(20,22);(8,10)(12,14)(19,20)(21,22);(11,12)(13,14)(16,18);(8,10)(12,14)(15,16)(17,18);(3,4,5,6)(7,8)(9,10)(11,12)(13,14);(1,2)

Conjugacy classes & character table

368 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72278177
maximal normal subgroup of	C₂×#72278177

GAP

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

gap> G := Group((7,8)(9,10)(20,22), (8,10)(12,14)(19,20)(21,22), (11,12)(13,14)(16,18), (8,10)(12,14)(15,16)(17,18), (3,4,5,6)(7,8)(9,10)(11,12)(13,14), (1,2));

C2×#72278057 #72297504