C₂²×#13142 #72313616

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#13142

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁴×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

6017 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#498375, C₂²×#495837, C₂²×#498116
maximal normal subgroup of	C₂²×#498375, C₂²×#495837, C₂²×#498116

GAP

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

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

C22×#13142 #72313616