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Cayley diagram

C3×#733⋊C2 #92329

order486
isoclinismisoclinic class of #733⋊C2
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×#733⋊C2

Splittings

Fitting, SylowC3×#733 ⋊ C2
derived#733 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3#733⋊C2
commutator, cosocle, hypocenter#733C2×C3
FrattiniC32C3×C3⋊D3
FittingC3×#733C2
socleC32#272
LCSG ▷ #733
UCSC1 ◁ C3

90 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)(6,13)(7,8)(10,12)(11,22)(14,25)(15,17)(16,24)(18,20)(19,26)(21,28)(23,29)(27,30);(4,16,13,7,17,15,8,6,24)(5,27,26,18,14,23,12,11,28)(9,21,22,10,29,25,20,19,30);(4,18,9,7,12,10,8,5,20)(6,11,29,16,27,19,17,14,21)(13,28,25,15,26,30,24,23,22)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#733⋊C2, C2×C3×#733⋊C2, #10624525⋊C2, …
maximal normal subgroup ofD3×#733⋊C2, C2×C3×#733⋊C2, #10624525⋊C2, …
maximal quotient ofC3×#733⋊C4, C2×C3×#733⋊C2, C3×#6858⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 174);