Finite Group Databasebeta

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Cayley diagram

C5×#638 #67179

order360
isoclinismisoclinic class of #634
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×#638

Splittings

FittingC22×C32×C5 ⋊ C2
HallC32×C5 ⋊ D4
SylowC32 ⋊ C5×D4

Subgroups

subgroupG/H
centerC2×C5C2×C3⋊D3
commutator, cosocleC2×C32C22×C5
FrattiniC2C2×C5×C3⋊D3
FittingC22×C32×C5C2
socleC2×C32×C5C22
hypercenterC22×C5C3⋊D3
hypocenterC32C5×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C22×C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 3 generators on 15 points:

(2,4)(8,9,10)(11,12,13,14,15);(1,2)(3,4)(6,7)(9,10);(5,6,7)

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10624163, #10624165, D5×#638, …
maximal normal subgroup of#10624163, #10624165, D5×#638, …
maximal quotient ofC5×#3750, C5×#3752, C5×#3753, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 109);