C₂²×#36159 #72324704

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

Aliases

direct product

C₂²×#36159

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

110537 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)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,22,23,24);(6,8)(19,23)(22,24);(10,13)(11,15);(3,4);(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,23)(18,22,20,24);(1,2)(3,4);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,22,23,24)

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#7630949, C₂²×#7716427, C₂²×#7675416
maximal normal subgroup of	C₂²×#7630949, C₂²×#7716427, C₂²×#7675416
maximal quotient of	C₂²×#612700, C₂²×#519708

GAP

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

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

C22×#36159 #72324704