C₅×#71890 #14288284

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

Aliases

direct product

C₅×#71890

Splittings

Hall	C₃×C₅ ⋊ #1608
Sylow	C₃ ⋊ C₅×#1608

Subgroups

	subgroup	G/H
center	C₂²×C₅	#953
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#543	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂⁴×C₃×C₅	D₄
hypercenter	C₅×#543	D₃
hypocenter	C₃	C₅×#1608
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#543

456 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 114 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)(6,7,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(9,10,12,14)(11,16,15,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(7,8)(10,13)(11,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)

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, 13982);