C₂²×#39434 #72299783

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

Aliases

direct product

C₂²×#39434

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

102773 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)(11,12)(13,15)(14,16)(19,24)(21,22);(9,11,13,16)(10,14,15,12)(17,18,20,23)(19,21,24,22);(11,16)(12,14)(17,19,20,24)(18,21,23,22);(6,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(3,4);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7880638, C₂²×#7696044
maximal normal subgroup of	C₂²×#7880638, C₂²×#7696044

GAP

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

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

C22×#39434 #72299783