C₅×#71189 #14293915

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

Aliases

direct product

C₅×#71189

Splittings

Hall	C₃×C₅ ⋊ #1719
Sylow	C₃ ⋊ C₅×#1719

Subgroups

	subgroup	G/H
center	C₂²×C₅	#940
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#486	C₂
socle	C₂²×C₃×C₅	#127
prime radical	C₃×C₅×#127	C₂²
hypercenter	C₅×#486	D₃
hypocenter	C₃	C₅×#1719
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#486

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

normal subgroups only

This group has 372 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) = 960

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

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

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, 19613);