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C5⋊((C19⋊C3)⋊D43) #37587799

order24510
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C19⋊C3)⋊D43)
unambiguous semidirect product(C19⋊C3)⋊D215

Splittings

Fitting, socle, derived, HallC5×C19×C43 ⋊ C2×C3
HallC5×C43×C19⋊C3 ⋊ C2
HallD4085 ⋊ C3
HallC43×C19⋊C3 ⋊ D5
HallC19×C43 ⋊ C3×D5
HallC5×C19⋊C3 ⋊ D43
HallC5×C43 ⋊ D19⋊C3
HallC5×C19 ⋊ C3×D43
HallC19⋊C3 ⋊ D215
SylowC43 ⋊ C5⋊(D19⋊C3)
SylowC19 ⋊ C3×D215
SylowC5 ⋊ (C19⋊C3)⋊D43

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C19×C43C2×C3
LCSG ▷ C5×C19×C43
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 4085

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

Polycyclic presentation:

2,3,5,19,43|c^a=c^4,d^a=d^18,e^a=e^42,d^b=d^7

Conjugacy classes & character table

972 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24510, 46);