C₂²×#14723 #72337053

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#14723

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

93989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#772779, C₂²×#1419356, C₂²×#612553, …
maximal normal subgroup of	C₂²×#772779, C₂²×#1419356, C₂²×#612553, …

GAP

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

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

C22×#14723 #72337053