Finite Group Databasebeta

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Cayley diagram

#6912⋊C2 #92334

order486
isoclinismisoclinic class of #6912⋊C2
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6912⋊C2

Splittings

Fitting, Sylow#6912 ⋊ C2
derived#733 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#733C2×C3
FrattiniC32C3×C3⋊D3
Fitting#6912C2
socleC3C3×#272
LCSG ▷ #733
UCSC1

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6912⋊C2, #10624750⋊C2, #13091123, …
maximal normal subgroup ofC2×#6912⋊C2, (#6880⋊C2)⋊C3, C3×#6912⋊C2
maximal quotient of#6912⋊C4, C2×#6912⋊C2, #10624592⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 179);