C₂²×#35129 #72324717

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

Aliases

direct product

C₂²×#35129

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

106365 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7726041, C₂²×#7653121, C₂²×#7631422
maximal normal subgroup of	C₂²×#7726041, C₂²×#7653121, C₂²×#7631422

GAP

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

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

C22×#35129 #72324717