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C5⋊(C5⋊(C17⋊(D13⋊C3))) #45684170

order33150
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC5⋊(C5⋊(C17⋊(D13⋊C3)))
unambiguous semidirect productC17⋊(C5⋊(C5⋊(D13⋊C3)))
unambiguous semidirect product(C13⋊C3)⋊(C5⋊D85)

Splittings

Fitting, socle, derived, HallC52×C13×C17 ⋊ C2×C3
HallC52×C17×C13⋊C3 ⋊ C2
HallC5⋊D1105 ⋊ C3
HallC52×C13⋊C3 ⋊ D17
HallC17×C13⋊C3 ⋊ C5⋊D5
HallC52×C17 ⋊ D13⋊C3
HallC52×C13 ⋊ C3×D17
HallC13×C17 ⋊ C3×C5⋊D5
HallC13⋊C3 ⋊ C5⋊D85
SylowC52 ⋊ C17⋊(D13⋊C3)
SylowC17 ⋊ C5⋊(C5⋊(D13⋊C3))
SylowC13 ⋊ C3×C5⋊D85

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C13×C17C2×C3
LCSG ▷ C52×C13×C17
UCSC1

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 5525

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(4,5)(7,8)(9,10)(12,16,17,13,14,18)(15,22,21,19,20,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40);(6,7,9,10,8)(11,12,15,14,18,21,20,22,23,17,16,19,13)(24,25,27,29,31,33,35,37,39,40,38,36,34,32,30,28,26);(1,2,4,5,3)

Polycyclic presentation:

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

Conjugacy classes & character table

1492 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33150, 52);