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Cayley diagram

#4177⋊C3 #92284

order486
isoclinismisoclinic class of #4177⋊C3
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect product#4177⋊C3

Splittings

Fitting, Sylow#6910 ⋊ C2
derivedC3×C9 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C9C2×C32
FrattiniC32C32×D3
Fitting#6910C2
socleC3C3×(C3⋊D3)⋊C3
LCSG ▷ C3×C9
UCSC1

76 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:

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

Polycyclic presentation:

2,3^5|d^3=f,d^a=d^2f^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(#4176⋊C2)⋊C3, C2×#4177⋊C3, #13091117, …
maximal normal subgroup of(#4176⋊C2)⋊C3, C2×#4177⋊C3, (#4169⋊C3)⋊C3, …
maximal quotient of#66324⋊C3, C2×#4177⋊C3, #13090756, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 129);