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Cayley diagram

C3×C5×SL(2,𝔽3) #67159

order360
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C4×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×SL(2,𝔽3)
unambiguous semidirect productQ8⋊(C32×C5)
unambiguous semidirect product(C3×Q8)⋊(C3×C5)
unambiguous semidirect product(C5×Q8)⋊C32
unambiguous semidirect product(C3×C5×Q8)⋊C3

Splittings

FittingC3×C5×Q8 ⋊ C3
derived, SylowQ8 ⋊ C32×C5
HallC5×Q8 ⋊ C32

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3×C5A4
commutator, cosocle, hypocenterQ8C32×C5
FrattiniC2C3×C5×A4
FittingC3×C5×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C5

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(C5⋊#226), C3⋊(C5⋊GL(2,𝔽3)), C3⋊#91184, …
maximal subgroup ofC3×C5×#226, C3×C5×GL(2,𝔽3), C3×C5⋊#226, …
maximal normal subgroup ofC3×C5×#226, C3×C5×GL(2,𝔽3), C3×C5⋊#226, …
maximal quotient ofC2×C3×C5×SL(2,𝔽3), C5×C9×SL(2,𝔽3), C5×C9⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 89);