C₂²×#12000 #72286975

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#12000

Subgroups

	subgroup	G/H
center	C₂³×C₄	#106
commutator	C₈	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂⁴	C₂×#106
prime radical	C₂⁵	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(1,2)(3,4)(14,19)(15,21)(20,27)(23,28);(13,14,17,20,16,21,24,28,18,19,26,27,25,15,22,23);(5,6,7,9,8,10,11,12)(13,15,16,19,18,21,25,14)(17,20,24,23,26,27,22,28)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#484147, C₂²×#483590, C₂²×#483463
maximal normal subgroup of	C₂²×#484147, C₂²×#483590, C₂²×#483463
maximal quotient of	C₂²×#135826

GAP

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

gap> G := Group((1,2), (3,4), (1,2)(3,4)(14,19)(15,21)(20,27)(23,28), (13,14,17,20,16,21,24,28,18,19,26,27,25,15,22,23), (5,6,7,9,8,10,11,12)(13,15,16,19,18,21,25,14)(17,20,24,23,26,27,22,28));

C22×#12000 #72286975