Finite Group Databasebeta

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C5×#84009 #14457669

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

Aliases

direct productC5×#84009

Splittings

FittingC3×C5×#576 ⋊ C2
HallC3×C5 ⋊ #3288
SylowC3 ⋊ C5×#3288

Subgroups

subgroupG/H
centerC2×C5D4×#67
commutator, cosocleC2×C3×C4C24×C5
FrattiniC2×C4C23×C5×D3
FittingC3×C5×#576C2
socleC2×C3×C5D42
hypercenterC5×#576D3
hypocenterC3C5×#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#576

928 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)(6,7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21);(9,10)(11,16)(12,15)(13,21)(14,17)(18,19)(20,24)(22,23);(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(7,8)(10,15)(12,13)(16,23)(17,21)(18,24)(19,22)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 183367);