C₂×#13106227 #72269237

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

Aliases

direct product

C₂×#13106227

Splittings

Sylow

C₅² ⋊ C₂²×#124

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
prime radical	C₂²×D₄×C₅⋊D₅	C₂
hypercenter	C₂²×D₄	D₅²
hypocenter	C₅²	C₂²×#124
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

5122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

248 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), (7,8,10,11,9)(17,19,20,24)(18,22,21,23), (1,2), (4,6)(13,15)(14,16)(17,18,20,21)(19,23,24,22), (3,4,5,6)(13,15)(14,16)(18,21)(22,23));

C2×#13106227 #72269237