C₅×#71326 #14290879

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#71326

Splittings

Fitting	C₃×C₅×#496 ⋊ C₂
Hall	C₃×C₅ ⋊ #1655
Sylow	C₃ ⋊ C₅×#1655

Subgroups

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#496	C₂
socle	C₂²×C₃×C₅	#120
hypercenter	C₅×#496	D₃
hypocenter	C₃	C₅×#1655
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#496

524 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 24
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(11,15)(13,16)(17,18)(19,24)(20,22)(21,23);(7,8)(9,10)(11,16)(12,14)(13,15)(19,23)(21,24);(10,13)(11,15)(14,16)(18,21)(19,23)(22,24)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 16577);