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Cayley diagram

C3×C5×C3⋊D3 #63208

order270
isoclinismisoclinic class of C3⋊D3
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×C3⋊D3
unambiguous semidirect productC3⋊(C3×C5×D3)

Splittings

Fitting, socle, HallC33×C5 ⋊ C2
derivedC32 ⋊ C2×C3×C5
SylowC33 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC3×C5C3⋊D3
commutator, cosocle, hypocenterC32C2×C3×C5
FrattiniC1G
Fitting, socleC33×C5C2
LCSG ▷ C32
UCSC1 ◁ C3×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD32⋊D15, #11762179, #11762180, …
maximal subgroup ofC3×C5×#157, C3×#4379, C5×C3⋊#157, …
maximal normal subgroup ofC3×C5×#157, C3×#4379, C5×C3⋊#157, …
maximal quotient ofC3×C5×C3⋊Dic3, C2×C3×C5×C3⋊D3, C3×C5×C3⋊D9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 26);