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C3⋊(((C31⋊C3)⋊C4)⋊C5) #19150069

order5580
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(((C31⋊C3)⋊C4)⋊C5)
unambiguous semidirect product((C31⋊C3)⋊C5)⋊Dic3
unambiguous semidirect product(C3⋊((C31⋊C3)⋊C4))⋊C5
unambiguous semidirect product(C3⋊(Dic31⋊C5))⋊C3

Splittings

derivedC3×C31 ⋊ C3×C4×C5
HallC3×(C31⋊C3)⋊C5 ⋊ C4
HallC3⋊((C31⋊C3)⋊C4) ⋊ C5
HallC3×C31⋊C3 ⋊ C4×C5
HallΓL(1,𝔽32) ⋊ C3×Dic3
SylowC31 ⋊ C3×C5×Dic3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊AGL(1,𝔽31)
commutator, hypocenterC3×C31C3×C4×C5
Fitting, socleC2×C3×C31C2×C3×C5
LCSG ▷ C3×C31
UCSC1 ◁ C2

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×C3⋊(((C31⋊C3)⋊C4)⋊C5), C3⋊(((C31⋊C3)⋊C5)⋊Dic5), C3⋊(((C31⋊C3)⋊C5)⋊AGL(1,𝔽5)), …
maximal normal subgroup ofC5×C3⋊(((C31⋊C3)⋊C4)⋊C5), C4×C3⋊AGL(1,𝔽31), D3×((C31⋊C3)⋊C4)⋊C5, …
maximal quotient ofC3⋊(((C31⋊C3)⋊C4)⋊C25), C5×C3⋊(((C31⋊C3)⋊C4)⋊C5), C3⋊(((C31⋊C3)⋊C5)⋊Dic5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5580, 58);