Finite Group Databasebeta

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Cayley diagram

C3×#1088 #66418

order324
isoclinismisoclinic class of #1088
automorphismAut(G) = D3×AΓL(1,𝔽9) (order 864) · Inn(G) = #157 · Out(G) = C22×D3
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#1088

Splittings

Fitting, SylowC3×H(3,𝔽3) ⋊ C4
derivedH(3,𝔽3) ⋊ C3×C4

Subgroups

subgroupG/H
center, socle, hypercenterC32#157
commutator, hypocenterH(3,𝔽3)C3×C4
FrattiniC3C3×#157
FittingC3×H(3,𝔽3)C4
prime radicalC3×#272C2
cosocle#272C2×C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C32

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3)(4,5)(6,17,12,10)(7,14)(8,11)(9,21,16,15)(13,20,19,18);(4,6,19)(5,18,9)(7,15,12)(8,13,21)(10,20,14)(11,16,17)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13049184, #13049186, #6249⋊D3
maximal subgroup ofC3×#6249, C3⋊#6249, D3×#1088, …
maximal normal subgroup ofC3×#6249, C3⋊#6249, D3×#1088, …
maximal quotient ofC3×#6188, C2×C3×#1088, C9×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 110);