C₂×#823456 #72302421

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#823456

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

62133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280357, C₂×#72280294, C₂×#72280449, …
maximal normal subgroup of	C₂×#72280357, C₂×#72280294, C₂×#72280449, …
maximal quotient of	C₂×#72278348, C₂²×#823456

GAP

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

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

C2×#823456 #72302421