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Cayley diagram

#6894⋊C2 #92263

order486
isoclinismisoclinic class of D9⋊H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6894⋊C2

Splittings

Fitting, Sylow#6894 ⋊ C2

Subgroups

subgroupG/H
centerC3C3×AGL(1,ℤ/9)
commutator, cosocleC3×C9C2×C32
Frattini, socleC32C32×D3
Fitting#6894C2
prime radicalC32×D9C3
hypercenterC32AGL(1,ℤ/9)
hypocenterC9C2×3−1+2
LCSG ▷ C3×C9 ▷ C9
UCSC1 ◁ C3 ◁ C32

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 18 points:

(2,5,6)(4,8,9)(11,13)(12,16)(14,15)(17,18);(1,6,9,3,5,4,7,2,8)(10,18,17,16,14,13,12,11,15)

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6894⋊C2, #13091212, #13091215, …
maximal normal subgroup ofC2×#6894⋊C2, #13091212, #13091215, …
maximal quotient of#6894⋊C4, C2×#6894⋊C2, #13090623, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 108);