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Cayley diagram

C3⋊(3−1+2⋊C4) #66408

order324
isoclinismisoclinic class of C3⋊AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(3−1+2⋊C4)

Splittings

derivedC3×C9 ⋊ C3×C4
SylowC3×3−1+2 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2C3⋊AGL(1,ℤ/9)
commutator, hypocenterC3×C9C3×C4
FrattiniC2×C3C3×C3⋊D3
FittingC2×C3×3−1+2C2
socleC2×C32C3×D3
prime radicalC2×C33D3
cosocleC2×C3×C9C2×C3
LCSG ▷ C3×C9
UCSC1 ◁ C2

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 16 points:

(2,3)(5,9,12,7,8,11)(6,10)(13,14,15,16);(1,2,3);(4,12,7,10,8,9,6,5,11)

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6189⋊C3, Dic3×AGL(1,ℤ/9), D3×3−1+2⋊C4, …
maximal normal subgroup of#6189⋊C3, Dic3×AGL(1,ℤ/9), D3×3−1+2⋊C4, …
maximal quotient ofC3⋊(3−1+2⋊C8), C2×C3⋊(3−1+2⋊C4), C3⋊#66316, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 100);