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C19⋊(D71⋊C5) #27212559

order13490
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC19⋊(D71⋊C5)
unambiguous semidirect product(C71⋊C5)⋊D19
unambiguous semidirect productD1349⋊C5

Splittings

Fitting, socle, derived, HallC19×C71 ⋊ C2×C5
HallC19×C71⋊C5 ⋊ C2
HallD1349 ⋊ C5
HallC71⋊C5 ⋊ D19
SylowC71 ⋊ C5×D19
SylowC19 ⋊ D71⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC19×C71C2×C5
LCSG ▷ C19×C71
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 90
transitive degreeμt(G) = 1349

A minimal faithful permutation representation — 2 generators on 90 points:

(2,6,7,25,30,3,4,12,14,49)(5,19,20,65,51,9,10,40,42,48)(8,35,36,37,54,16,17,61,43,63)(11,47,38,39,24,22,23,18,64,26)(13,50,29,46,70,27,28,52,53,69)(15,57,58,55,56,32,33,59,31,62)(21,34,68,41,71,44,45,60,66,67)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90);(1,2,5,11,24,7,15,34,56,61,20,43,35,55,25,38,16,6,13,29,23,48,50,60,70,30,54,71,65,64,58,57,53,44,19,41,66,10,21,46,33,59,39,42,67,63,49,69,68,28,51,47,52,27,4,8,18,14,31,17,37,40,36,62,45,32,12,26,22,9,3)(72,73,75,77,79,81,83,85,87,89,90,88,86,84,82,80,78,76,74)

Polycyclic presentation:

2,5,19,71|c^a=c^18,d^a=d^70,d^b=d^54

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD19×D71⋊C5, C2×C19⋊(D71⋊C5)
maximal normal subgroup ofD19×D71⋊C5, C2×C19⋊(D71⋊C5)
maximal quotient ofC19⋊(Dic71⋊C5), C2×C19⋊(D71⋊C5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13490, 12);