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D85⋊(D13⋊C3) #27210508

order13260
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD85⋊(D13⋊C3)
unambiguous semidirect product(C5⋊(D13⋊C3))⋊D17
unambiguous semidirect product(C17⋊(D13⋊C3))⋊D5
unambiguous semidirect product(D65⋊D17)⋊C3

Splittings

Fitting, socle, derived, HallC5×C13×C17 ⋊ C22×C3
HallD65⋊D17 ⋊ C3
HallC5×C17×C13⋊C3 ⋊ C22
HallC17×C13⋊C3 ⋊ C2×D5
HallC13×C17 ⋊ C2×C3×D5
HallC5×C13⋊C3 ⋊ C2×D17
HallC5×C17 ⋊ C2×D13⋊C3
HallC5×C13 ⋊ C2×C3×D17
HallC13⋊C3 ⋊ D5×D17
SylowC17 ⋊ D5×D13⋊C3
SylowC13 ⋊ C3×D5×D17
SylowC5 ⋊ D17×D13⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C13×C17C22×C3
LCSG ▷ C5×C13×C17
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 1105

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,26,29,27,24)(20,25,22,30,21,28)(32,33)(34,35);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,20)(21,23)(22,26)(24,25)(27,29)(28,30)(31,32,34,35,33)

Polycyclic presentation:

2^2,3,5,13,17|e^a=e^12,f^a=f^16,d^b=d^4,e^b=e^12,e^c=e^3

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13260, 58);