Finite Group Databasebeta

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C5×#64085 #13088059

order1440
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×#64085

Splittings

FittingC32×C42×C5 ⋊ C2
HallC32×C5 ⋊ #126
SylowC32 ⋊ C5×#126

Subgroups

subgroupG/H
centerC22×C5C22×C3⋊D3
commutator, cosocleC22×C32C23×C5
FrattiniC22C22×C5×C3⋊D3
FittingC32×C42×C5C2
socleC22×C32×C5C23
prime radicalC22×C5×C3⋊D3C22
hypercenterC42×C5C3⋊D3
hypocenterC32C5×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C42×C5

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5)(9,10,11)(13,17)(16,19)(20,22,23,27)(21,25,24,26);(6,7,8)(12,14,15,18)(13,16,17,19)(21,24)(25,26);(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21)(22,26)(23,24)(25,27)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4376);