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C5⋊((C19⋊C3)⋊(D19⋊C3)) #44570541

order32490
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC5⋊((C19⋊C3)⋊(D19⋊C3))
unambiguous semidirect product(C19⋊C3)⋊(C5⋊(D19⋊C3))
unambiguous semidirect product(C5⋊#14524206)⋊C3
unambiguous semidirect product(C5⋊#14524207)⋊C3
unambiguous semidirect product(C5⋊((C19⋊C3)⋊D19))⋊C3

Splittings

Fitting, socle, derived, HallC5×C192 ⋊ C2×C32
HallC5×C19⋊C32 ⋊ C2
HallC5⋊(C19⋊D19) ⋊ C32
HallC19⋊C32 ⋊ D5
SylowC192 ⋊ C32×D5
SylowC5 ⋊ (C19⋊C3)⋊(D19⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C192C2×C32
LCSG ▷ C5×C192
UCSC1

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 285

A minimal faithful permutation representation — 2 generators on 43 points:

(2,3)(4,5)(7,11,12,8,9,13)(10,19,16,14,15,21)(17,22,24,23,20,18)(25,26)(27,29)(28,36)(30,32)(31,37)(33,35)(34,42)(38,39)(40,41);(1,2,4,5,3)(6,7,10,16,22,19,23,9,13,18,24,12,11,17,15,20,21,14,8)(26,28,31)(27,30,32)(29,34,35)(33,38,40)(36,39,43)(37,42,41)

Polycyclic presentation:

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

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 125);