Finite Group Databasebeta

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A4×AGL(1,𝔽5)×AGL(1,ℤ/9) #72247693

order12960
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productA4×AGL(1,𝔽5)×AGL(1,ℤ/9)

Splittings

FittingC22×C5×3−1+2 ⋊ C2×C3×C4
derivedC22×C5×C9 ⋊ C2×C32×C4
SylowC5 ⋊ C4×A4×AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC22×C5×C9C2×C32×C4
FrattiniC3C3×D3×A4×AGL(1,𝔽5)
FittingC22×C5×3−1+2C2×C3×C4
socleC22×C3×C5C32×C4×D3
prime radicalD5×A4×AGL(1,ℤ/9)C2
cosocleC22×C9×D5C22×C32
LCSG ▷ C22×C5×C9
UCSC1

1118 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:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×S4×AGL(1,ℤ/9)
maximal normal subgroup ofAGL(1,𝔽5)×S4×AGL(1,ℤ/9)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4,5)(6,10,14,8,7,9,13,12,11)(16,18,17), (1,2,5,3,4)(7,11,12,9,10,14)(8,13)(15,18)(16,17));