C₂×#4306434 #72279414

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#4306434

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40663 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×#2397, C₂×#72280318, C₂×#72279903, …
maximal normal subgroup of	C₂×D₄×#2397, C₂×#72280318, C₂×#72279903, …

GAP

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

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

C2×#4306434 #72279414