C₂×#13106188 #72269246

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#13106188

Splittings

Fitting	C₂²×C₅²×D₄ ⋊ C₂²
Sylow	C₅² ⋊ C₂×#535

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅²
commutator, cosocle	C₂²×C₅²	C₂⁵
Frattini	C₂²	C₂³×D₅²
Fitting	C₂²×C₅²×D₄	C₂²
socle	C₂³×C₅²	C₂⁴
hypercenter	C₂²×D₄	D₅²
hypocenter	C₅²	C₂×#535
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

4290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

GAP

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

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

C2×#13106188 #72269246