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Cayley diagram

C2×C4×AGL(1,ℤ/9) #88706

order432
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = AGL(1,ℤ/9) · Out(G) = #120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C2×C3×C4)
unambiguous semidirect product(C2×C4×D9)⋊C3

Splittings

FittingC2×C4×3−1+2 ⋊ C2
derivedC9 ⋊ C22×C3×C4
Sylow3−1+2 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,ℤ/9)
commutator, hypocenterC9C22×C3×C4
FrattiniC2×C3C22×C3×D3
FittingC2×C4×3−1+2C2
socleC22×C3C2×C3×D3
prime radicalC22×AGL(1,ℤ/9)C2
cosocleC2×C9C23×C3
LCSG ▷ C9
UCSC1 ◁ C2×C4

170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,ℤ/9))⋊C8, C42×AGL(1,ℤ/9), #63434⋊C3, …
maximal normal subgroup of(C2×AGL(1,ℤ/9))⋊C8, C42×AGL(1,ℤ/9), #63434⋊C3, …
maximal quotient ofC4×3−1+2⋊Q8, C42×AGL(1,ℤ/9), #63434⋊C3, …
automorphism group ofC9⋊C16, C5×Dic9, C2×C5×D9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 353);