Finite Group Databasebeta

← Prev (#10623630) Next (#10623632) →

Dic9⋊AGL(1,𝔽5) #10623631

order720
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productDic9⋊AGL(1,𝔽5)

Splittings

HallC5×C9 ⋊ C4⋊C4
SylowC9 ⋊ C4⋊AGL(1,𝔽5)
SylowC5 ⋊ Dic9⋊C4

Subgroups

subgroupG/H
center, hypercenterC2D9×AGL(1,𝔽5)
commutator, FittingC2×C5×C9C2×C4
FrattiniC2×C3D3×AGL(1,𝔽5)
socleC2×C3×C5C4×D3
prime radicalC2×C3×D5C2×D3
cosocleC2×C9×D5C22
hypocenterC5×C9C4⋊C4
LCSG ▷ C2×C5×C9 ▷ C5×C9
UCSC1 ◁ C2

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

Polycyclic presentation:

2^4,3,5,3|a^2=c,b^2=d,e^3=g,b^a=bd,f^a=f^2,e^b=e^2g^2,g^b=g^2,f^c=f^4

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×Dic18, (C9⋊AGL(1,𝔽5))⋊Q8, #13085172, …
maximal normal subgroup ofAGL(1,𝔽5)×Dic18, (C9⋊AGL(1,𝔽5))⋊Q8, #13085172, …
maximal quotient of#13083914, #13083915, #13083922, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 100);