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C5⋊(C41⋊(D19⋊C3)) #37552597

order23370
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C41⋊(D19⋊C3))
unambiguous semidirect productC41⋊(C5⋊(D19⋊C3))
unambiguous semidirect product(C19⋊C3)⋊D205
unambiguous semidirect productD3895⋊C3

Splittings

Fitting, socle, derived, HallC5×C19×C41 ⋊ C2×C3
HallC5×C41×C19⋊C3 ⋊ C2
HallD3895 ⋊ C3
HallC41×C19⋊C3 ⋊ D5
HallC19×C41 ⋊ C3×D5
HallC5×C19⋊C3 ⋊ D41
HallC5×C41 ⋊ D19⋊C3
HallC5×C19 ⋊ C3×D41
HallC19⋊C3 ⋊ D205
SylowC41 ⋊ C5⋊(D19⋊C3)
SylowC19 ⋊ C3×D205
SylowC5 ⋊ C41⋊(D19⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C19×C41C2×C3
LCSG ▷ C5×C19×C41
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 65
transitive degreeμt(G) = 3895

A minimal faithful permutation representation — 2 generators on 65 points:

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

Polycyclic presentation:

2,3,5,19,41|c^a=c^4,d^a=d^18,e^a=e^40,d^b=d^7

Conjugacy classes & character table

927 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23370, 33);