C₂²×#10729 #72315090

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10729

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂⁴	C₂×#95
prime radical	C₂⁷	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

45125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#359476, C₂×#72237188, C₂×#72239650, …
maximal normal subgroup of	C₂²×#359476, C₂×#72237188, C₂×#72239650, …

GAP

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

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

C22×#10729 #72315090