C₂²×#42474 #72324592

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

Aliases

direct product

C₂²×#42474

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

115601 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)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,22,20,23);(6,8)(9,10,12,14)(11,13,15,16)(17,19)(18,22)(20,21)(23,24);(11,15)(13,16)(17,18)(19,22)(20,23)(21,24);(3,4);(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22);(1,2)(3,4);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,22)(21,24)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7916101, C₂²×#7655557, C₂²×#7966358, …
maximal normal subgroup of	C₂²×#7916101, C₂²×#7655557, C₂²×#7966358, …
maximal quotient of	C₂²×#686267, C₂²×#772784

GAP

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

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

C22×#42474 #72324592