C₂²×#39128 #72324648

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

Aliases

direct product

C₂²×#39128

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

100961 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)(11,16)(12,13)(14,15)(17,18)(19,24)(20,22)(21,23);(9,11,14,12)(10,13,15,16)(17,19,21,20)(18,22,23,24);(18,23)(22,24);(3,4);(6,8)(9,12,14,11)(10,13,15,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4);(5,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,24,23,22)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7677211
maximal normal subgroup of	C₂²×#7677211

GAP

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

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

C22×#39128 #72324648