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Cayley diagram

C3⋊#4178 #92327

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

Aliases

unambiguous semidirect productC3⋊#4178

Splittings

Fitting, SylowC3×#732 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC3C3⋊((C3⋊D3)⋊C3)
commutator, cosocle, hypocenterC3×H(3,𝔽3)C2×C3
FrattiniC32C3×C3⋊D3
FittingC3×#732C2
socleC32(C3⋊D3)⋊C3
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C3

96 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)(4,12,18,8,11,9,7,5,19)(6,29,21,24,10,30,16,23,17,13,22,28,15,20,25,26,14,27);(1,2,3);(4,6,13)(5,10,20)(7,16,26)(8,15,24)(9,17,27)(11,22,29)(12,14,23)(18,25,30)(19,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#4178, #4179⋊D3, #11757757, …
maximal normal subgroup ofD3×#4178, #4179⋊D3, #11757757, …
maximal quotient ofC3⋊#66325, C2×C3⋊#4178, C3⋊#92160, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 172);