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Cayley diagram

C3×#131⋊C3 #63990

order288
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C23×D3
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×#131⋊C3
unambiguous semidirect product#131⋊C32
unambiguous semidirect product(C3×#131)⋊C3

Splittings

FittingC3×#131 ⋊ C3
Sylow#131 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3×C8A4
commutator, hypocenterQ8C32×C4
FrattiniC4C2×C3×A4
FittingC3×#131C3
socleC2×C3C4×A4
prime radicalC3×#41⋊C3C2
cosocle#41C2×C32
LCSG ▷ Q8
UCSC1 ◁ C3×C8

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,7,13,2,6,20,3,10)(4,19,27,8,17,31,11,24,14,18,28,21,15,32,25,22,5,29,26,9,16,30,12,23);(1,14,6,16)(2,21,10,23)(3,25,13,27)(4,17,15,5)(7,29,20,31)(8,24,22,9)(11,28,26,12)(18,32,30,19)(33,34,35)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11920597, #11920598, #11920599, …
maximal subgroup ofC3×#4628, C3⋊#4628, C3⋊(#504⋊C3), …
maximal normal subgroup ofC3×#4628, C3⋊#4628, C3⋊(#504⋊C3), …
maximal quotient ofC3×C8×SL(2,𝔽3), C2×C3×#131⋊C3, C9×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 638);