C₂²×#39116 #72324662

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

Aliases

direct product

C₂²×#39116

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

95845 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)(9,10,14,12)(11,15,13,16)(17,18,20,22)(19,23,24,21);(9,11,14,13)(10,15,12,16)(17,18,20,22)(19,23,24,21);(10,12)(15,16);(3,4);(6,8)(9,12,14,10)(11,15,13,16)(17,19,20,24)(18,21,22,23);(1,2)(3,4);(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,18)(19,21)(20,22)(23,24)

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

C22×#39116 #72324662