Finite Group Databasebeta

← Prev (#37558211) Next (#37558213) →

C5⋊((C61⋊C3)⋊D13) #37558212

order23790
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C61⋊C3)⋊D13)
unambiguous semidirect product(C61⋊C3)⋊D65

Splittings

Fitting, socle, derived, HallC5×C13×C61 ⋊ C2×C3
HallC5×C13×C61⋊C3 ⋊ C2
HallD3965 ⋊ C3
HallC13×C61⋊C3 ⋊ D5
HallC5×C61⋊C3 ⋊ D13
HallC13×C61 ⋊ C3×D5
HallC5×C61 ⋊ C3×D13
HallC61⋊C3 ⋊ D65
HallC5×C13 ⋊ D61⋊C3
SylowC61 ⋊ C3×D65
SylowC13 ⋊ C5⋊(D61⋊C3)
SylowC5 ⋊ (C61⋊C3)⋊D13

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C13×C61C2×C3
LCSG ▷ C5×C13×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 3965

A minimal faithful permutation representation — 2 generators on 79 points:

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

Polycyclic presentation:

2,3,5,13,61|c^a=c^4,d^a=d^12,e^a=e^60,e^b=e^47

Conjugacy classes & character table

752 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23790, 57);