C₅×#84126 #14425747

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₅×#84126

Splittings

Fitting	C₃×C₅×#585 ⋊ C₂
Hall	C₃×C₅ ⋊ #3027
Sylow	C₃ ⋊ C₅×#3027

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#953
commutator, cosocle	C₂³×C₃	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#585	C₂
socle	C₂×C₃×C₅	C₂×#120
hypercenter	C₅×#585	D₃
hypocenter	C₃	C₅×#3027
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#585

1544 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 151445);