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C5⋊((C13⋊C3)⋊D49) #34062052

order19110
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C13⋊C3)⋊D49)
unambiguous semidirect product(C13⋊C3)⋊D245

Splittings

Fitting, derived, HallC5×C13×C49 ⋊ C2×C3
HallC5×C13⋊C3×C49 ⋊ C2
HallD3185 ⋊ C3
HallC13⋊C3×C49 ⋊ D5
HallC13×C49 ⋊ C3×D5
HallC5×C49 ⋊ D13⋊C3
HallC5×C13⋊C3 ⋊ D49
HallC5×C13 ⋊ C3×D49
SylowC49 ⋊ C5⋊(D13⋊C3)
HallC13⋊C3 ⋊ D245
SylowC13 ⋊ C3×D245
SylowC5 ⋊ (C13⋊C3)⋊D49

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC5×C13×C49C2×C3
FrattiniC7C5⋊((C13⋊C3)⋊D7)
socleC5×C7×C13C3×D7
LCSG ▷ C5×C13×C49
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 3185

A minimal faithful permutation representation — 2 generators on 67 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,17)(16,20)(18,23)(19,26)(21,29)(22,28)(24,33)(25,36)(27,38)(30,42)(31,41)(32,40)(34,35)(37,47)(39,49)(43,50)(44,52)(45,51)(46,55)(48,57)(53,58)(54,59)(56,61)(60,62)(64,65)(66,67);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,50,25,59,45,47,31,20,39,18,62,33,57,22,29,27,40,56,34,53,17,42,19,52,46,55,44,26,30,15,58,35,61,32,38,21,28,48,24,60,23,49,16,41,37,51,54,36,43)(63,64,66,67,65)

Polycyclic presentation:

2,3,5,7,13,7|d^7=f^6,c^a=c^4,d^a=d^6f,e^a=e^12,f^a=f^6,e^b=e^3

Conjugacy classes & character table

862 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD13⋊C3×D245, D49×C5⋊(D13⋊C3), D5×(C13⋊C3)⋊D49, …
maximal normal subgroup ofD13⋊C3×D245, D49×C5⋊(D13⋊C3), D5×(C13⋊C3)⋊D49, …
maximal quotient of#50303359, C2×C5⋊((C13⋊C3)⋊D49)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19110, 45);