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C3⋊(C3⋊(C5⋊(D31⋊C5))) #28358469

order13950
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C5⋊(D31⋊C5)))
unambiguous semidirect productC5⋊(C3⋊(C3⋊(D31⋊C5)))
unambiguous semidirect productΓL(1,𝔽32)⋊(C3⋊D15)
unambiguous semidirect product(C3⋊D465)⋊C5

Splittings

Fitting, socle, derivedC32×C5×C31 ⋊ C2×C5
HallC32×C5×ΓL(1,𝔽32) ⋊ C2
HallC5×ΓL(1,𝔽32) ⋊ C3⋊D3
HallC32×C31 ⋊ C5×D5
SylowC31 ⋊ C5×C3⋊D15
SylowC32 ⋊ C5⋊(D31⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C5×C31C2×C5
LCSG ▷ C32×C5×C31
UCSC1

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,5,3^2,5,31|c^a=c^2,d^a=d^2,e^a=e^4,f^a=f^30,f^b=f^16

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C3⋊(C5⋊(D31⋊C5)), D15×C3⋊(D31⋊C5), C3⋊D15×D31⋊C5, …
maximal normal subgroup ofD3×C3⋊(C5⋊(D31⋊C5)), D15×C3⋊(D31⋊C5), C3⋊D15×D31⋊C5, …
maximal quotient ofC3⋊(C3⋊(ΓL(1,𝔽32)⋊Dic5)), C2×C3⋊(C3⋊(C5⋊(D31⋊C5)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13950, 112);