C₂²×#45455 #72324821

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

Aliases

direct product

C₂²×#45455

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

108049 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7712649, C₂²×#7768137, C₂²×#7763623, …
maximal normal subgroup of	C₂²×#7712649, C₂²×#7768137, C₂²×#7763623, …
maximal quotient of	C₂²×#856513

GAP

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

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

C22×#45455 #72324821