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C5×C3⋊D663 #34091061

order19890
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D663
unambiguous semidirect productC3⋊(C5×D663)
unambiguous semidirect productC13⋊(C5×C3⋊D51)
unambiguous semidirect productC17⋊(C5×C3⋊D39)

Splittings

Fitting, socle, HallC32×C5×C13×C17 ⋊ C2
derived, HallC32×C13×C17 ⋊ C2×C5
HallC5×C13×C17 ⋊ C3⋊D3
HallC32×C5×C17 ⋊ D13
HallC32×C5×C13 ⋊ D17
HallC13×C17 ⋊ C5×C3⋊D3
HallC32×C17 ⋊ C5×D13
HallC32×C13 ⋊ C5×D17
HallC5×C17 ⋊ C3⋊D39
HallC5×C13 ⋊ C3⋊D51
HallC32×C5 ⋊ D221
SylowC17 ⋊ C5×C3⋊D39
SylowC13 ⋊ C5×C3⋊D51
SylowC32 ⋊ C5×D221

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D663
commutator, cosocle, hypocenterC32×C13×C17C2×C5
FrattiniC1G
Fitting, socleC32×C5×C13×C17C2
LCSG ▷ C32×C13×C17
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 9945

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

4980 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19890, 46);