Finite Group Databasebeta

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Cayley diagram

C32×#272 #92385

order486
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC32×#272

Splittings

Fitting, SylowC32×H(3,𝔽3) ⋊ C2
derivedH(3,𝔽3) ⋊ C2×C32

Subgroups

subgroupG/H
center, socle, hypercenterC33C3⋊D3
commutator, cosocle, hypocenterH(3,𝔽3)C2×C32
FrattiniC3C32×C3⋊D3
FittingC32×H(3,𝔽3)C2
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C33

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

derived subgroup of(((C3⋊D3)⋊C3)⋊C2)⋊(C3⋊D3), #14519128, #14519132, …
maximal subgroup ofC32×#1088, C3×C3⋊#1088, C3⋊(C3⋊#1088), …
maximal normal subgroup ofC32×#1088, C3×C3⋊#1088, C3⋊(C3⋊#1088), …
maximal quotient ofC32×#1084, C2×C32×#272, C32×#4181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 230);