C₂²×#36147 #72324901

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

Aliases

direct product

C₂²×#36147

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

111057 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)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22);(6,8)(10,13)(14,15)(17,18)(19,23)(20,24)(21,22);(18,22)(20,23);(3,4);(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4);(9,10)(11,15)(12,13)(14,16)(17,20)(18,19)(21,23)(22,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7670211, C₂²×#7854849, C₂²×#7685422
maximal normal subgroup of	C₂²×#7670211, C₂²×#7854849, C₂²×#7685422

GAP

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

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

C22×#36147 #72324901