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Cayley diagram

#3673⋊C3 #88508

order432
isoclinismisoclinic class of #3672⋊C3
automorphismAut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = #611⋊C3 · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#3673⋊C3

Splittings

FittingD4×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ D8

Subgroups

subgroupG/H
centerC2#611⋊C3
commutator, cosocleC4×C9C22×C3
FrattiniC3×C4C2×C3×D3
FittingD4×3−1+2C2
socleC2×C3C3×#67
hypercenterD4AGL(1,ℤ/9)
hypocenterC9C3×D8
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C4 ◁ D4

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

2^2,3,2^2,3^2|d^2=e,f^3=g^2,b^a=bd,d^a=de,f^a=f^2g,g^a=g^2,d^b=de,f^c=fg

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×AGL(1,ℤ/9), #63473⋊C3, #63476⋊C3, …
maximal normal subgroup ofD8×AGL(1,ℤ/9), #63473⋊C3, #63476⋊C3, …
maximal quotient of#63366⋊C3, #63369⋊C3, #63385⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 155);