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Cayley diagram

C3×#859 #63751

order288
isoclinismisoclinic class of #226
automorphismAut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#859

Splittings

derivedSL(2,𝔽3) ⋊ C3×C4

Subgroups

subgroupG/H
center, socle, hypercenterC22×C3S4
commutator, hypocenterSL(2,𝔽3)C3×C4
FrattiniC22C3×S4
FittingC2×C3×Q8D3
prime radicalC2×C3×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C2×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3

63 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)(5,6,12,7,8,11,10,9)(13,14,15);(6,9,12)(7,10,11)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593191, #10593193, #10593195, …
maximal normal subgroup of#10593191, #10593193, #10593195, …
maximal quotient ofC3×SL(2,𝔽3)⋊C8, C2×C3×#859, C9×#859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 399);