Finite Group Databasebeta

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Cayley diagram

C3⋊(#733⋊C2) #92343

order486
isoclinismisoclinic class of C3⋊(#733⋊C2)
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect productC3⋊(#733⋊C2)

Splittings

Fitting, derived, SylowC3×#733 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×#733C2
FrattiniC32C3⋊(C3⋊D3)
socleC32#272
LCSG ▷ C3×#733
UCSC1

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 4 generators on 30 points:

(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);(1,2,3)

Polycyclic presentation:

2,3^5|c^3=f^2,d^3=f^2,b^a=b^2,c^a=c^2f,d^a=d^2f,e^a=ef^2,f^a=f^2,d^c=de,e^c=ef

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#733⋊C2, C2×C3⋊(#733⋊C2), #10624478⋊C2, …
maximal normal subgroup ofD3×#733⋊C2, C2×C3⋊(#733⋊C2), #10624478⋊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, 188);