C₂²×#14786 #72337048

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂²×#14786

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

76293 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)(10,12)(17,18,20,22)(19,21,24,23);(5,6,7,8)(10,12)(14,16)(17,19)(18,21)(20,24)(22,23);(6,8)(9,10,11,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23);(3,4);(5,6)(7,8)(9,10)(11,12)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#759718, C₂²×#846737, C₂²×#612485, …
maximal normal subgroup of	C₂²×#759718, C₂²×#846737, C₂²×#612485, …

GAP

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

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

C22×#14786 #72337048