C₂²×#38023 #72324672

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#38023

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

112133 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7721725, C₂²×#7675441, C₂²×#7653121
maximal normal subgroup of	C₂²×#7721725, C₂²×#7675441, C₂²×#7653121
maximal quotient of	C₂²×#647717

GAP

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

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

C22×#38023 #72324672