C₂²×#35084 #72324729

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#35084

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

113181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7628282, C₂²×#7652921
maximal normal subgroup of	C₂²×#7628282, C₂²×#7652921

GAP

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

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

C22×#35084 #72324729