C₂²×#35554 #72337189

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

Aliases

direct product

C₂²×#35554

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

129685 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7643002, C₂²×#7646026, C₂²×#7631310, …
maximal normal subgroup of	C₂²×#7643002, C₂²×#7646026, C₂²×#7631310, …
maximal quotient of	C₂²×#515363

GAP

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

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

C22×#35554 #72337189