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C5×AGL(1,𝔽9)⋊C4 #13088392

order1440
isoclinismisoclinic class of AΓL(1,𝔽9)
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×AGL(1,𝔽9)⋊C4
unambiguous semidirect productAGL(1,𝔽9)⋊(C4×C5)

Splittings

HallC32×C5 ⋊ #106
SylowC32 ⋊ C5×#106

Subgroups

subgroupG/H
center, hypercenterC2×C5AΓL(1,𝔽9)
commutator#157C2×C4×C5
FrattiniC2C5×AΓL(1,𝔽9)
Fitting, socleC2×C32×C5SD16
prime radicalC2×C5×C3⋊D3D4
cosocleC2×#157C22×C5
hypocenterC32C5×#106
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C5

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(6,11,13)(7,10,12,9,14,8)(15,16,17,18);(7,8,9,13,10,12,14,11)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×C5×AΓL(1,𝔽9), C3×C5×AGL(1,𝔽9)⋊C4
maximal normal subgroup ofC4×C5×AΓL(1,𝔽9), C3×C5×AGL(1,𝔽9)⋊C4
maximal quotient ofC3×C5×AGL(1,𝔽9)⋊C4

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4709);