C₂²×#13775 #72296047

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁴×C₄²

Aliases

direct product

C₂²×#13775

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁴×C₄²
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

176213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#763814, C₂²×#576607, C₂²×#521311, …
maximal normal subgroup of	C₂²×#763814, C₂²×#576607, C₂²×#521311, …

GAP

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

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

C22×#13775 #72296047