C₂²×#59190 #72324824

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

Aliases

direct product

C₂²×#59190

Subgroups

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

101269 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7763623
maximal normal subgroup of	C₂²×#7763623
maximal quotient of	C₂²×#647923

GAP

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

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

C22×#59190 #72324824