Finite Group Databasebeta

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AGL(1,𝔽5)×S4×AGL(1,ℤ/9) #72269576

order25920
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productAGL(1,𝔽5)×S4×AGL(1,ℤ/9)

Splittings

FittingC22×C5×3−1+2 ⋊ C2×C4×D3
derivedC5×C9×A4 ⋊ C22×C3×C4
SylowC5 ⋊ C4×S4×AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC5×C9×A4C22×C3×C4
FrattiniC3C3×D3×AGL(1,𝔽5)×S4
FittingC22×C5×3−1+2C2×C4×D3
socleC22×C3×C5C3×C4×D32
prime radicalD5×S4×AGL(1,ℤ/9)C2
cosocleC9×D5×A4C23×C3
LCSG ▷ C5×C9×A4
UCSC1

3156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

holomorph ofC22×C5×C9

GAP

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

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