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Cayley diagram

3−1+2⋊C16 #88384

order432
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×C4×AGL(1,ℤ/9) (order 432) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×C4
homologyAbelianization (H1) = C3×C16 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product3−1+2⋊C16
unambiguous semidirect product(C9⋊C16)⋊C3

Splittings

derivedC9 ⋊ C3×C16
Sylow3−1+2 ⋊ C16

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,ℤ/9)
commutator, hypocenterC9C3×C16
FrattiniC3×C8C3×D3
FittingC8×3−1+2C2
socleC2×C3C3×C3⋊C8
prime radicalC2×C32C3⋊C8
cosocleC8×C9C2×C3
LCSG ▷ C9
UCSC1 ◁ C8

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(2,6,9,4,5,8)(3,7)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(1,9,4,7,5,6,3,2,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×AGL(1,ℤ/9), #63357⋊C3, C2×3−1+2⋊C16, …
maximal normal subgroup ofC16×AGL(1,ℤ/9), #63357⋊C3, C2×3−1+2⋊C16, …
maximal quotient of3−1+2⋊C32, C2×3−1+2⋊C16, #13046312, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 31);