C₅×#68347 #14284329

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#68347

Splittings

Fitting	C₃×C₅×#329 ⋊ C₂
Hall	C₃×C₅ ⋊ #1533
Sylow	C₃ ⋊ C₅×#1533

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#329	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×D₃×D₄	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#329	D₃
hypocenter	C₃	C₅×#1533
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#329

496 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 480

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

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 10027);