C₂²×#14262 #72293779

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#14262

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

170805 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#616969, C₂²×#523233, C₂²×#686982, …
maximal normal subgroup of	C₂²×#616969, C₂²×#523233, C₂²×#686982, …

GAP

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

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

C22×#14262 #72293779