C₅×#68491 #14289055

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

Aliases

direct product

C₅×#68491

Splittings

Fitting	C₃×C₅×#333 ⋊ C₂
Hall	C₃×C₅ ⋊ #1624
Sylow	C₃ ⋊ C₅×#1624

Subgroups

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#333	C₂
socle	C₂²×C₃×C₅	#120
hypercenter	C₅×#333	D₃
hypocenter	C₃	C₅×#1624
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#333

480 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 960

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

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

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, 14753);