C₂²×#34982 #72296001

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

Aliases

direct product

C₂²×#34982

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

204053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7630944, C₂²×#7670206, C₂²×#7628174
maximal normal subgroup of	C₂²×#7630944, C₂²×#7670206, C₂²×#7628174
maximal quotient of	C₂²×#612469, C₂²×#517417, C₂²×#516395

GAP

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

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

C22×#34982 #72296001