C₂²×#38028 #72325824

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

Aliases

direct product

C₂²×#38028

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

109357 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)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,23)(21,24);(10,13)(14,15)(17,19)(18,21)(20,24)(22,23);(6,8)(11,16)(14,15)(17,19)(18,22)(20,24)(21,23);(3,4);(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24);(1,2)(3,4);(9,11)(10,14)(12,16)(13,15)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7722013, C₂²×#7675441, C₂²×#7736715
maximal normal subgroup of	C₂²×#7722013, C₂²×#7675441, C₂²×#7736715

GAP

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

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

C22×#38028 #72325824