C₅×#68998 #14308950

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

Aliases

direct product

C₅×#68998

Splittings

Hall	C₃×C₅ ⋊ #2028
Sylow	C₃ ⋊ C₅×#2028

Subgroups

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#343	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₅×#343	D₃
hypocenter	C₃	C₅×#2028
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#343

492 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 134 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)(6,7)(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25);(9,11,12,16)(10,13,14,15)(20,29)(22,31)(23,28)(26,30);(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 34648);