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Cayley diagram

#4175⋊C3 #92282

order486
isoclinismisoclinic class of #4175⋊C3
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect product#4175⋊C3

Splittings

Fitting, Sylow#6909 ⋊ C2
derivedC33 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC33C2×C32
FrattiniC32C32×D3
Fitting#6909C2
socleC3C3×(C3⋊D3)⋊C3
LCSG ▷ C33
UCSC1

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(#4174⋊C2)⋊C3, C2×#4175⋊C3, #92258⋊C3, …
maximal normal subgroup of(#4174⋊C2)⋊C3, C2×#4175⋊C3, #92258⋊C3, …
maximal quotient of#66322⋊C3, C2×#4175⋊C3, #13090750, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 127);