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Cayley diagram

C3×D13⋊C3 #6618

order234
isoclinismisoclinic class of D13⋊C3
automorphismAut(G) = D3×AGL(1,𝔽13) (order 936) · Inn(G) = D13⋊C3 · Out(G) = C2×D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×D13⋊C3
unambiguous semidirect productD13⋊C32
unambiguous semidirect product(C13⋊C3)⋊(C2×C3)
unambiguous semidirect product(C3×D13)⋊C3

Splittings

Fitting, socleC3×C13 ⋊ C2×C3
derived, SylowC13 ⋊ C2×C32
HallC3×C13⋊C3 ⋊ C2
HallD13 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3D13⋊C3
commutator, cosocle, hypocenterC13C2×C32
FrattiniC1G
Fitting, socleC3×C13C2×C3
LCSG ▷ C13
UCSC1 ◁ C3

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 39

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽13), C3⋊AGL(1,𝔽13), D3×D13⋊C3, …
maximal normal subgroup ofC3×AGL(1,𝔽13), C3⋊AGL(1,𝔽13), D3×D13⋊C3, …
maximal quotient ofC3×Dic13⋊C3, C2×C3×D13⋊C3, C9×D13⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 7);