C₂²×#35776 #72324644

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

Aliases

direct product

C₂²×#35776

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

97333 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)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24);(6,8)(9,11,14,13)(10,15,16,12)(19,23)(21,24);(6,8)(10,16)(11,13)(17,18,20,22)(19,24,23,21);(3,4);(9,12)(10,11)(13,16)(14,15)(17,19,20,23)(18,21,22,24);(1,2)(3,4);(5,6)(7,8)(9,13,14,11)(10,15,16,12)(17,18)(19,24)(20,22)(21,23)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7694192
maximal normal subgroup of	C₂²×#7694192

GAP

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

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

C22×#35776 #72324644