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C5⋊(C5⋊((C19⋊C3)⋊D13)) #49150305

order37050
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC5⋊(C5⋊((C19⋊C3)⋊D13))
unambiguous semidirect product(C19⋊C3)⋊(C5⋊D65)

Splittings

Fitting, socle, derived, HallC52×C13×C19 ⋊ C2×C3
HallC52×C13×C19⋊C3 ⋊ C2
HallC5⋊D1235 ⋊ C3
HallC52×C19⋊C3 ⋊ D13
HallC13×C19⋊C3 ⋊ C5⋊D5
HallC52×C19 ⋊ C3×D13
HallC52×C13 ⋊ D19⋊C3
HallC13×C19 ⋊ C3×C5⋊D5
HallC19⋊C3 ⋊ C5⋊D65
SylowC52 ⋊ (C19⋊C3)⋊D13
SylowC19 ⋊ C3×C5⋊D65
SylowC13 ⋊ C5⋊(C5⋊(D19⋊C3))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C13×C19C2×C3
LCSG ▷ C52×C13×C19
UCSC1

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 6175

A minimal faithful permutation representation — 3 generators on 42 points:

(2,3)(4,5)(7,8)(9,10)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,29,30,26,27,31)(28,37,34,32,33,39)(35,40,42,41,38,36);(6,7,9,10,8)(11,12,14,16,18,20,22,23,21,19,17,15,13)(24,25,28,34,40,37,41,27,31,36,42,30,29,35,33,38,39,32,26);(1,2,4,5,3)

Polycyclic presentation:

2,3,5^2,13,19|c^a=c^4,d^a=d^4,e^a=e^12,f^a=f^18,f^b=f^7

Conjugacy classes & character table

1467 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37050, 131);