C₂²×#36318 #72296533

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

Aliases

direct product

C₂²×#36318

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

108893 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)(11,15)(14,16)(18,21)(19,23);(11,15)(14,16);(10,13)(11,15)(18,21)(19,23);(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7850315
maximal normal subgroup of	C₂²×#7850315
maximal quotient of	C₂²×#1159493

GAP

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

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

C22×#36318 #72296533