C₂²×#38906 #72324676

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

Aliases

direct product

C₂²×#38906

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

115909 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7726041, C₂²×#7664581, C₂²×#7733277, …
maximal normal subgroup of	C₂²×#7726041, C₂²×#7664581, C₂²×#7733277, …
maximal quotient of	C₂²×#773162

GAP

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

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

C22×#38906 #72324676