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Cayley diagram

C3×C4×SL(2,𝔽3) #63985

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×C4×SL(2,𝔽3)
unambiguous semidirect product(C4×Q8)⋊C32
unambiguous semidirect product(C3×C4×Q8)⋊C3

Splittings

FittingC3×C4×Q8 ⋊ C3
derivedQ8 ⋊ C32×C4
SylowC4×Q8 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4A4
commutator, hypocenterQ8C32×C4
FrattiniC22C2×C3×A4
FittingC3×C4×Q8C3
socleC22×C3C2×A4
prime radicalC2×C3×SL(2,𝔽3)C2
cosocleC2×Q8C2×C32
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C4

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11920501, #11920502, #11920503, …
maximal subgroup ofC3×SL(2,𝔽3)⋊C8, C3⋊(SL(2,𝔽3)⋊C8), C3⋊C8×SL(2,𝔽3), …
maximal normal subgroup ofC3×SL(2,𝔽3)⋊C8, C3⋊(SL(2,𝔽3)⋊C8), C3⋊C8×SL(2,𝔽3), …
maximal quotient ofC3×C8×SL(2,𝔽3), C2×C3×C4×SL(2,𝔽3), C4×C9×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 633);