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Cayley diagram

C3×C3⋊SL(2,ℤ/4) #88971

order432
isoclinismisoclinic class of C3⋊S4
automorphismAut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C3⋊SL(2,ℤ/4)
unambiguous semidirect productC3⋊(C3×SL(2,ℤ/4))
unambiguous semidirect productA4⋊(C3×Dic3)

Splittings

derivedC3×A4 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3C3⋊S4
commutator, hypocenterC3×A4C3×C4
FrattiniC2C3×C3⋊S4
Fitting, socleC23×C32D3
prime radicalC2×C32×A4C2
cosocleC2×C3×A4C2×C3
LCSG ▷ C3×A4
UCSC1 ◁ C2×C3

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64204, C3×Dic3×S4, C3×D3×SL(2,ℤ/4), …
maximal normal subgroup ofC3×#64204, C3×Dic3×S4, C3×D3×SL(2,ℤ/4), …
maximal quotient ofC3×C3⋊(A4⋊C8), C3×C3⋊#859, C3×C3⋊GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 618);