Finite Group Databasebeta

← Prev (#92338) Next (#92340) →

Cayley diagram

C3⋊(C3⋊D27) #92339

order486
isoclinismisoclinic class of C3⋊(C3⋊D27)
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C33

Aliases

unambiguous semidirect productC3⋊(C3⋊D27)
unambiguous semidirect productC27⋊(C3⋊D3)

Splittings

Fitting, derived, SylowC32×C27 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC32×C27C2
FrattiniC9C3⋊(C3⋊D3)
socleC33D9
LCSG ▷ C32×C27
UCSC1

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 243

A minimal faithful permutation representation — 4 generators on 33 points:

(2,3)(5,6)(8,11)(9,15)(10,18)(12,13)(14,24)(16,17)(19,26)(20,25)(21,31)(22,30)(23,29)(27,28)(32,33);(4,5,6);(1,2,3);(7,26,20,27,31,29,9,8,30,10,33,24,16,13,12,17,14,32,18,22,11,15,23,21,28,25,19)

Polycyclic presentation:

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

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊D3×D27, D3×C3⋊D27, C2×C3⋊(C3⋊D27), …
maximal normal subgroup ofC3⋊D3×D27, D3×C3⋊D27, C2×C3⋊(C3⋊D27), …
maximal quotient ofC3⋊(C3⋊Dic27), C2×C3⋊(C3⋊D27), C3⋊(C9⋊D27), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 184);