C₂²×#45105 #72299810

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

Aliases

direct product

C₂²×#45105

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

99233 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7696044, C₂²×#7878926, C₂²×#7968016, …
maximal normal subgroup of	C₂²×#7696044, C₂²×#7878926, C₂²×#7968016, …

GAP

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

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

C22×#45105 #72299810