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Cayley diagram

#611⋊C3 #6225

order216
isoclinismisoclinic class of 3−1+2⋊Q8
automorphismAut(G) = C22×AGL(1,ℤ/9) (order 216) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C2
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#611⋊C3

Splittings

FittingC22×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ D4

Subgroups

subgroupG/H
centerC2C2×AGL(1,ℤ/9)
commutator, cosocleC2×C9C22×C3
Frattini, socleC2×C3C2×C3×D3
FittingC22×3−1+2C2
hypercenterC22AGL(1,ℤ/9)
hypocenterC9C3×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22

53 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

2^2,3,2,3^2|e^3=f^2,b^a=bd,e^a=e^2f,f^a=f^2,e^c=ef

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#3697⋊C3, D4×AGL(1,ℤ/9), #3699⋊C3, …
maximal normal subgroup of#3697⋊C3, D4×AGL(1,ℤ/9), #3699⋊C3, …
maximal quotient of(3−1+2⋊C4)⋊C4, (C2×AGL(1,ℤ/9))⋊C4, #3672⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 62);