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Cayley diagram

D9×AGL(1,𝔽5) #67109

order360
isoclinismisoclinic class of D9×AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,ℤ/9) (order 1080) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productD9×AGL(1,𝔽5)
unambiguous semidirect product(C9×AGL(1,𝔽5))⋊C2
unambiguous semidirect product(C9⋊AGL(1,𝔽5))⋊C2

Splittings

Fitting, derived, HallC5×C9 ⋊ C2×C4
SylowC9 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×D9

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterC5×C9C2×C4
FrattiniC3D3×AGL(1,𝔽5)
socleC3×C5C4×D3
prime radicalD5×D9C2
cosocleC9×D5C22
LCSG ▷ C5×C9
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D9×AGL(1,𝔽5), AGL(1,𝔽5)×D27, C3×D9×AGL(1,𝔽5), …
maximal normal subgroup ofC2×D9×AGL(1,𝔽5), C3×D9×AGL(1,𝔽5), AGL(1,𝔽5)×AGL(1,ℤ/9), …
maximal quotient ofD9×#170, #10623627, #10623628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 39);