C₂²×#13143 #72310245

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

Aliases

direct product

C₂²×#13143

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5525 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)(10,14,17,21)(11,18)(12,19)(15,20,24,22)(16,23);(5,6,7,8)(10,15,17,24)(11,16)(14,20,21,22)(18,23);(6,8)(9,10,16,21,12,15,23,20,13,17,11,14,19,24,18,22);(1,2);(3,4)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#498120, C₂²×#498375, C₂²×#495843
maximal normal subgroup of	C₂²×#498120, C₂²×#498375, C₂²×#495843

GAP

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

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

C22×#13143 #72310245