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Cayley diagram

#6864⋊C2 #92170

order486
isoclinismisoclinic class of D27⋊C3
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6864⋊C2

Splittings

Fitting, Sylow#6864 ⋊ C2
derivedC27 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC3D27⋊C3
commutator, hypocenterC27C2×C9
FrattiniC3×C9C3×D3
Fitting#6864C2
socleC32C3×D9
prime radicalC3×D27C3
cosocleC3×C27C2×C3
LCSG ▷ C27
UCSC1 ◁ C3

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 36 points:

(1,5,9,3,8,4,6,2,7)(11,22,25,14,15,24)(12,18)(13,21)(16,27,26,17,28,35)(19,20)(23,36,34,30,33,29)(31,32);(10,30,24,31,35,28,12,11,34,13,36,22,19,17,16,20,15,33,21,29,14,18,27,26,32,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6864⋊C2, #10624462⋊C2, #10624500⋊C2, …
maximal normal subgroup ofC2×#6864⋊C2, #10624462⋊C2, C3×#6864⋊C2, …
maximal quotient of#6864⋊C4, C2×#6864⋊C2, #10624443⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 15);