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Cayley diagram

C3×#41⋊C3 #3814

order144
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×D3×S4 (order 288) · Inn(G) = A4 · Out(G) = C22×D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×#41⋊C3
unambiguous semidirect product#41⋊C32
unambiguous semidirect product(C3×#41)⋊C3

Splittings

FittingC3×#41 ⋊ C3
Sylow#41 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3×C4A4
commutator, cosocle, hypocenterQ8C2×C32
FrattiniC2C2×C3×A4
FittingC3×#41C3
socleC2×C3C2×A4
LCSG ▷ Q8
UCSC1 ◁ C3×C4

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(1,8,5,2)(3,16,11,6,4,15,10,7,9,14,12,13);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)(17,18,19)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10593223, #10593224, #10593225, …
maximal subgroup ofC3×GU(2,𝔽3), C3⋊GU(2,𝔽3), C3⋊(#131⋊C3), …
maximal normal subgroup ofC3×GU(2,𝔽3), C3⋊GU(2,𝔽3), C3⋊(#131⋊C3), …
maximal quotient ofC3×C4×SL(2,𝔽3), C2×C3×#41⋊C3, C9×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 157);