Finite Group Databasebeta

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Cayley diagram

C3×C3⋊(C3⋊(C3⋊D3)) #92414

order486
isoclinismisoclinic class of C3⋊(C3⋊(C3⋊D3))
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C36

Aliases

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

Splittings

Fitting, socle, SylowC35 ⋊ C2
derivedC34 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3C3⋊(C3⋊(C3⋊D3))
commutator, cosocle, hypocenterC34C2×C3
FrattiniC1G
Fitting, socleC35C2
LCSG ▷ C34
UCSC1 ◁ C3

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14519527, #14519528, C3⋊#10620211
maximal subgroup ofC3×#66472, C3⋊#66472, C3×D3×C3⋊(C3⋊D3), …
maximal normal subgroup ofC3×#66472, C3⋊#66472, C3×D3×C3⋊(C3⋊D3), …
maximal quotient ofC3×C3⋊(C3⋊(C3⋊Dic3)), C2×C3×C3⋊(C3⋊(C3⋊D3)), C9×C3⋊(C3⋊(C3⋊D3)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 259);