C₂²×#38084 #72335670

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#38084

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

110765 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)(19,22,24,23);(6,7)(10,13)(11,15)(18,20)(22,23);(10,13)(14,16)(17,19)(18,22)(20,23)(21,24);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,23,24,22);(3,4);(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,22,24,23);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7645254, C₂²×#7684117, C₂²×#7967933
maximal normal subgroup of	C₂²×#7645254, C₂²×#7684117, C₂²×#7967933
maximal quotient of	C₂²×#524465, C₂²×#649110

GAP

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

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

C22×#38084 #72335670