C₂²×#39722 #72324563

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

Aliases

direct product

C₂²×#39722

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

93913 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)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,22,23,24);(6,8)(9,10,12,14)(11,15,16,13)(17,19)(18,22)(20,24)(21,23);(9,10)(11,13)(12,14)(15,16);(3,4);(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,22,23,24);(1,2)(3,4);(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7708684
maximal normal subgroup of	C₂²×#7708684

GAP

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

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

C22×#39722 #72324563