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C17⋊(C25⋊(D13⋊C3)) #45684142

order33150
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC17⋊(C25⋊(D13⋊C3))
unambiguous semidirect productC25⋊(C17⋊(D13⋊C3))
unambiguous semidirect product(C13⋊C3)⋊D425
unambiguous semidirect productD5525⋊C3

Splittings

Fitting, derived, HallC13×C17×C25 ⋊ C2×C3
HallC17×C25×C13⋊C3 ⋊ C2
HallD5525 ⋊ C3
HallC25×C13⋊C3 ⋊ D17
HallC17×C13⋊C3 ⋊ D25
HallC17×C25 ⋊ D13⋊C3
HallC13×C25 ⋊ C3×D17
HallC13×C17 ⋊ C3×D25
HallC13⋊C3 ⋊ D425
SylowC25 ⋊ C17⋊(D13⋊C3)
SylowC17 ⋊ C25⋊(D13⋊C3)
SylowC13 ⋊ C3×D425

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC13×C17×C25C2×C3
FrattiniC5C5⋊(C17⋊(D13⋊C3))
socleC5×C13×C17C3×D5
LCSG ▷ C13×C17×C25
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 5525

A minimal faithful permutation representation — 2 generators on 55 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(32,34)(33,37)(35,40)(36,43)(38,46)(39,45)(41,42)(44,50)(47,51)(48,52)(49,54)(53,55);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,17,19,21,23,25,27,29,30,28,26,24,22,20,18,16)(31,45,55,48,34,37,51,49,40,43,46,44,42,41,50,38,36,35,54,47,33,32,52,53,39)

Polycyclic presentation:

2,3,5,13,17,5|c^5=f^4,c^a=c^4f,d^a=d^12,e^a=e^16,f^a=f^4,d^b=d^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, 24);