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C5×PSU(3,𝔽2)⋊C4 #13088391

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

Aliases

direct productC5×PSU(3,𝔽2)⋊C4
unambiguous semidirect productPSU(3,𝔽2)⋊(C4×C5)

Splittings

HallC32×C5 ⋊ #103
SylowC32 ⋊ C5×#103

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×#103
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C5

100 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)(15,17,18,16);(7,8,9,13,10,12,14,11)(15,16,18,17)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×(PSU(3,𝔽2)⋊C3)⋊C4, C3×C5×PSU(3,𝔽2)⋊C4, C4×C5×AΓL(1,𝔽9)
maximal normal subgroup ofC3×C5×PSU(3,𝔽2)⋊C4, C4×C5×AΓL(1,𝔽9)
maximal quotient ofC3×C5×PSU(3,𝔽2)⋊C4

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4708);