C₂²×#13065 #72313859

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13065

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495895, C₂²×#498116, C₂²×D₄×#360
maximal normal subgroup of	C₂²×#495895, C₂²×#498116, C₂²×D₄×#360

GAP

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

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

C22×#13065 #72313859