C₅×#69180 #14318031

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#69180

Splittings

Hall	C₃×C₅ ⋊ #2228
Sylow	C₃ ⋊ C₅×#2228

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#357	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×D₈	C₂²
hypercenter	C₅×#357	D₃
hypocenter	C₃	C₅×#2228
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#357

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,7,8)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(7,8)(9,11,12,16)(10,13,14,15)(18,22)(23,30)(24,31)(25,29);(9,10,12,14)(11,15,16,13)(17,18,19,22)(20,23,26,30)(21,29,27,25)(24,32,31,28)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 43729);