C₅×#68092 #14293798

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅×#68092

Splittings

Fitting	C₃×C₅×#322 ⋊ C₂
Hall	C₃×C₅ ⋊ #1717
Sylow	C₃ ⋊ C₅×#1717

Subgroups

	subgroup	G/H
center	C₂²×C₅	#874
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	#127
prime radical	C₅×#4695	C₂
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1717
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322

416 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 122 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)(6,7,8)(10,14)(13,16)(17,19,21,18,22,23,20,24);(9,10,11,13,12,14,15,16)(17,18,21,23,22,24,20,19);(7,8)(10,13)(11,15)(14,16)(18,19)(20,21)(23,24)

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 19496);