Finite Group Databasebeta

← Prev (#92322) Next (#92324) →

Cayley diagram

C3×#4184 #92323

order486
isoclinismisoclinic class of #4184
automorphismAut(G) = C2×(#4176⋊C2)⋊C3 (order 1944) · Inn(G) = #4184 · Out(G) = C22×C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×#4184

Splittings

Fitting, SylowC3×#731 ⋊ C2
derived#731 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3#4184
commutator, cosocle, hypocenter#731C2×C3
FrattiniC32C3×C3⋊D3
FittingC3×#731C2
socleC32#272
LCSG ▷ #731
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(5,9)(6,12)(7,8)(10,20)(11,22)(13,25)(14,15)(16,24)(17,19)(18,28)(21,30)(23,26)(27,29);(4,16,12,8,15,14,7,6,24)(5,13,28,11,27,21,17,10,23)(9,26,20,19,30,29,22,18,25);(4,5,9)(6,13,26)(7,17,22)(8,11,19)(10,18,15)(12,23,25)(14,28,20)(16,27,30)(21,29,24)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#4176⋊C2, D3×#4184, #4184⋊D3, …
maximal normal subgroup ofC3×#4176⋊C2, D3×#4184, #4184⋊D3, …
maximal quotient ofC3×#66331, C2×C3×#4184, C3×#92199, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 168);