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Cayley diagram

C3⋊#4177 #92325

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

Aliases

unambiguous semidirect productC3⋊#4177

Splittings

Fitting, SylowC3×#731 ⋊ C2
derivedC32×C9 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×C9C2×C3
FrattiniC32C3×C3⋊D3
FittingC3×#731C2
socleC32(C3⋊D3)⋊C3
LCSG ▷ C32×C9
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 81

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#4177, #11757738, #4184⋊D3, …
maximal normal subgroup ofD3×#4177, #11757738, #4184⋊D3, …
maximal quotient ofC3⋊#66324, C2×C3⋊#4177, C3⋊#92161, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 170);