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C5×#80305 #14425741

order1920
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25×C4

Aliases

direct productC5×#80305

Splittings

FittingC3×C5×B(4,𝔽2) ⋊ C2
HallC3×C5 ⋊ #3027
SylowC3 ⋊ C5×#3027

Subgroups

subgroupG/H
centerC2×C5D3×#120
commutator, cosocleC23×C3C24×C5
FrattiniC23C23×C5×D3
FittingC3×C5×B(4,𝔽2)C2
socleC2×C3×C5C2×#120
hypercenterC5×B(4,𝔽2)D3
hypocenterC3C5×#3027
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C23×C5 ◁ C5×B(4,𝔽2)

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

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

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

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, 151439);