C₂×#72250287 #72334632

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72250287

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7964 conjugacy classes of subgroups; only the 12312 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298627
maximal normal subgroup of	C₂×#72298627

GAP

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

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

C2×#72250287 #72334632