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Cayley diagram

C3×C22⋊SL(2,𝔽3) #64338

order288
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC3×C22⋊SL(2,𝔽3)
unambiguous semidirect product(C22×C3)⋊SL(2,𝔽3)

Splittings

FittingC22×C3×Q8 ⋊ C3
derived, SylowC22×Q8 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3C22⋊A4
commutator, cosocle, hypocenterC22×Q8C32
FrattiniC2C3×C22⋊A4
FittingC22×C3×Q8C3
socleC23×C3A4
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C3

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊#5930, C3⋊#5931, C3⋊#68001, …
maximal subgroup ofC3⋊(#395⋊C3), C3⋊(#393⋊C3), C3×#393⋊C3, …
maximal normal subgroup ofC3⋊(#395⋊C3), C3⋊(#393⋊C3), C3×#393⋊C3, …
maximal quotient ofC3×#543⋊C3, C3×Q8⋊SL(2,𝔽3), C2×C3×C22⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 986);