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Cayley diagram

C3⋊#4176 #92324

order486
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊#4176

Splittings

Fitting, SylowC3×#731 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC3C3⋊((C3⋊D3)⋊C3)
commutator, cosocle, hypocenterC3×H(3,𝔽3)C2×C3
FrattiniC32C3×C3⋊D3
FittingC3×#731C2
socleC32(C3⋊D3)⋊C3
prime radicalC3⋊#272C3
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C3

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 30 points:

(2,3)(4,10,27,7,21,8,15,5,17)(6,18,16,23,20,30,13,22,19,11,9,26,14,29,24,25,12,28);(1,2,3);(4,6,11)(5,9,18)(7,14,23)(8,16,26)(10,20,29)(12,22,21)(13,25,15)(17,24,30)(19,28,27)

Polycyclic presentation:

2,3^5|b^3=f,c^a=c^2,d^a=d^2,e^a=e^2f^2,c^b=ce,e^c=ef^2

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#4176, #4177⋊D3, #4184⋊D3, …
maximal normal subgroup ofD3×#4176, #4177⋊D3, #4184⋊D3, …
maximal quotient ofC3⋊#66323, C2×C3⋊#4176, C3⋊#92162, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 169);