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AGL(1,𝔽5)×C27⋊C3 #13108170

order1620
automorphism?
homologyAbelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×C27⋊C3
unambiguous semidirect product(AGL(1,𝔽5)×C27)⋊C3

Splittings

Fitting, HallC5×C27⋊C3 ⋊ C4
SylowC5 ⋊ C4×C27⋊C3

Subgroups

subgroupG/H
center, FrattiniC9C32×AGL(1,𝔽5)
commutator, socleC3×C5C3×C4×C9
FittingC5×C27⋊C3C4
prime radicalC32×D5C2×C9
cosocleC9×D5C2×C32
hypercenterC27⋊C3AGL(1,𝔽5)
hypocenterC5C4×C27⋊C3
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C9 ◁ C27⋊C3

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 135

A minimal faithful permutation representation — 2 generators on 32 points:

(2,3,4,5)(6,7,10,8,12,18,14,21,26,9,13,17,15,22,25,23,29,31,16,11,19,24,20,27,30,28,32);(1,2,3,5,4)(7,11,13)(10,17,19)(12,20,22)(18,25,27)(21,28,29)(26,31,32)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#6875, AGL(1,𝔽5)×D27⋊C3
maximal normal subgroup ofAGL(1,𝔽5)×#6875, AGL(1,𝔽5)×D27⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 76);