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Cayley diagram

Dic3×SL(2,𝔽3) #63761

order288
isoclinismisoclinic class of C3⋊(#41⋊C3)
automorphismAut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×A4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productDic3×SL(2,𝔽3)
unambiguous semidirect productC3⋊(C4×SL(2,𝔽3))
unambiguous semidirect product(Q8×Dic3)⋊C3

Splittings

derivedC3×Q8 ⋊ C3×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC22D3×A4
commutator, hypocenterC3×Q8C3×C4
Fitting, cosocleC2×C3×Q8C2×C3
socleC22×C3C2×A4
prime radicalC2×C3×SL(2,𝔽3)C2
LCSG ▷ C3×Q8
UCSC1 ◁ C22

67 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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×#226, #10593193, #10593195, …
maximal normal subgroup ofDic3×#226, #10593193, #10593195, …
maximal quotient ofC3⋊C8×SL(2,𝔽3), C2×Dic3×SL(2,𝔽3), Dic9⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 409);