C₂²×#38022 #72324661

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#38022

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

118685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(6,8)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24);(10,13)(11,15)(20,24)(22,23);(18,19)(22,23);(3,4);(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22);(1,2)(3,4);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7677211, C₂²×#7664581, C₂²×#7721725, …
maximal normal subgroup of	C₂²×#7677211, C₂²×#7664581, C₂²×#7721725, …
maximal quotient of	C₂²×#524461, C₂²×#772795, C₂²×#647716

GAP

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

gap> G := Group((6,8)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24), (10,13)(11,15)(20,24)(22,23), (18,19)(22,23), (3,4), (6,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22), (1,2)(3,4), (5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23));

C22×#38022 #72324661