Finite Group Databasebeta

← Prev (#36348645) Next (#36348647) →

C5⋊(C11⋊(D67⋊C3)) #36348646

order22110
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C11⋊(D67⋊C3))
unambiguous semidirect productC11⋊(C5⋊(D67⋊C3))
unambiguous semidirect product(C67⋊C3)⋊D55
unambiguous semidirect productD3685⋊C3

Splittings

Fitting, socle, derived, HallC5×C11×C67 ⋊ C2×C3
HallC5×C11×C67⋊C3 ⋊ C2
HallD3685 ⋊ C3
HallC11×C67⋊C3 ⋊ D5
HallC5×C67⋊C3 ⋊ D11
HallC11×C67 ⋊ C3×D5
HallC5×C67 ⋊ C3×D11
HallC67⋊C3 ⋊ D55
SylowC67 ⋊ C3×D55
HallC5×C11 ⋊ D67⋊C3
SylowC11 ⋊ C5⋊(D67⋊C3)
SylowC5 ⋊ C11⋊(D67⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C11×C67C2×C3
LCSG ▷ C5×C11×C67
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 3685

A minimal faithful permutation representation — 2 generators on 83 points:

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

Polycyclic presentation:

2,3,5,11,67|c^a=c^4,d^a=d^10,e^a=e^66,e^b=e^37

Conjugacy classes & character table

692 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22110, 42);