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Cayley diagram

C5×D13⋊C3 #87608

order390
isoclinismisoclinic class of D13⋊C3
automorphismAut(G) = C4×AGL(1,𝔽13) (order 624) · Inn(G) = D13⋊C3 · Out(G) = C2×C4
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D13⋊C3
unambiguous semidirect productD13⋊(C3×C5)
unambiguous semidirect product(C13⋊C3)⋊(C2×C5)
unambiguous semidirect product(C5×D13)⋊C3

Splittings

Fitting, socle, HallC5×C13 ⋊ C2×C3
derived, SylowC13 ⋊ C2×C3×C5
HallC5×C13⋊C3 ⋊ C2
HallC5×D13 ⋊ C3
HallC13⋊C3 ⋊ C2×C5
HallD13 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5D13⋊C3
commutator, cosocle, hypocenterC13C2×C3×C5
FrattiniC1G
Fitting, socleC5×C13C2×C3
LCSG ▷ C13
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 65

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(7,11,12,8,9,13)(10,17,16,14,15,18);(6,7,10,9,13,16,15,17,18,12,11,14,8)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽13), #63118⋊C3, D5×D13⋊C3, …
maximal normal subgroup ofC5×AGL(1,𝔽13), #63118⋊C3, D5×D13⋊C3, …
maximal quotient ofC5×Dic13⋊C3, C2×C5×D13⋊C3, C5×D13⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(390, 1);