C₂×C₅×#66049 #72269240

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×C₅×#66049

Splittings

Fitting	C₂×C₅²×#127 ⋊ C₂
Sylow	C₅² ⋊ C₂×#535

Subgroups

	subgroup	G/H
center	C₂³×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₅
Fitting	C₂×C₅²×#127	C₂
socle	C₂³×C₅²	C₂⁴
hypercenter	C₂×C₅×#127	D₅
hypocenter	C₅	C₂×C₅×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#127

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:

(7,8,9,10,11)(12,13,15,16,14)(17,19,20,23)(18,22,21,24);(13,14)(15,16)(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

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

GAP

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

gap> G := Group((7,8,9,10,11)(12,13,15,16,14)(17,19,20,23)(18,22,21,24), (13,14)(15,16)(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×C5×#66049 #72269240