C₂²×#34986 #72324727

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

Aliases

direct product

C₂²×#34986

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

110869 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7716427, C₂²×#7628174
maximal normal subgroup of	C₂²×#7716427, C₂²×#7628174
maximal quotient of	C₂²×#612517, C₂²×#612509

GAP

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

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

C22×#34986 #72324727