C₂²×C₂²⋊#338 #72315098

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×C₂²⋊#338

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂⁴	C₂×#95
prime radical	C₂⁷	Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

44437 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)(11,18)(15,23)(19,24)(20,21);(6,8)(9,10,12,16,14,17,13,22)(11,19,20,15,18,24,21,23);(6,8)(9,11,13,20,14,18,12,21)(10,15,16,19,17,23,22,24);(1,2);(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#359476, C₂×#72239647, C₂×#72239648
maximal normal subgroup of	C₂²×#359476, C₂×#72239647, C₂×#72239648

GAP

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

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

C22×C22⋊#338 #72315098