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Cayley diagram

#4183⋊C3 #92331

order486
isoclinismisoclinic class of #4183⋊C3
automorphismAut(G) = (#4174⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#4183⋊C3

Splittings

Fitting, Sylow#6909 ⋊ C2
derivedAΣL(1,𝔽27) ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterAΣL(1,𝔽27)C2×C3
FrattiniC32C3×C3⋊D3
Fitting#6909C2
socleC3C3×#272
LCSG ▷ AΣL(1,𝔽27)
UCSC1

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(#4174⋊C2)⋊C3, C2×#4183⋊C3, #13091107, …
maximal normal subgroup of(#4174⋊C2)⋊C3, C2×#4183⋊C3, C3×#4183⋊C3, …
maximal quotient of#66330⋊C3, C2×#4183⋊C3, #10624590⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 176);