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D5×AΓL(1,𝔽9) #13089544

order1440
isoclinismisoclinic class of D5×AΓL(1,𝔽9)
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productD5×AΓL(1,𝔽9)
unambiguous semidirect product(D5×AGL(1,𝔽9))⋊C2
unambiguous semidirect product(C5×AΓL(1,𝔽9))⋊C2
unambiguous semidirect product#10624311⋊C2
unambiguous semidirect product(PSU(3,𝔽2)⋊D5)⋊C2
unambiguous semidirect product(AGL(1,𝔽9)⋊D5)⋊C2
unambiguous semidirect product(D5×PSU(3,𝔽2))⋊C2

Splittings

Fitting, socle, HallC32×C5 ⋊ C2×SD16
SylowC32 ⋊ D5×SD16
SylowC5 ⋊ C2×AΓL(1,𝔽9)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC5×#157C23
Fitting, socle, hypocenterC32×C5C2×SD16
prime radicalD5×GO+(4,𝔽2)C2
LCSG ▷ C5×#157 ▷ C5×C3⋊D3 ▷ C32×C5
UCSC1

168 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 45

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×AGL(2,𝔽3), AGL(1,𝔽5)×AΓL(1,𝔽9), C3×D5×AΓL(1,𝔽9), …
maximal normal subgroup ofAGL(1,𝔽5)×AΓL(1,𝔽9), C3×D5×AΓL(1,𝔽9), C2×D5×AΓL(1,𝔽9)
maximal quotient ofC3×D5×AΓL(1,𝔽9), C2×D5×AΓL(1,𝔽9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5861);