C₂²×#37670 #72299805

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

Aliases

direct product

C₂²×#37670

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

94849 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)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22);(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24);(9,11,13,16)(10,15,12,14)(17,20,22,23)(18,19,21,24);(6,8)(9,12,13,10)(11,14,16,15)(17,21,22,18)(19,23,24,20);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,21)(22,24);(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×#7708684, C₂²×#7694192
maximal normal subgroup of	C₂²×#7696044, C₂²×#7708684, C₂²×#7694192

GAP

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

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

C22×#37670 #72299805