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C5×C3⋊D651 #34086293

order19530
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D651
unambiguous semidirect productC3⋊(C5×D651)
unambiguous semidirect productC7⋊(C5×C3⋊D93)

Splittings

Fitting, socle, HallC32×C5×C7×C31 ⋊ C2
derived, HallC32×C7×C31 ⋊ C2×C5
HallC32×C5×C31 ⋊ D7
HallC5×C7×C31 ⋊ C3⋊D3
HallC32×C5×C7 ⋊ D31
HallC32×C31 ⋊ C5×D7
HallC7×C31 ⋊ C5×C3⋊D3
HallC5×C31 ⋊ C3⋊D21
HallC32×C7 ⋊ C5×D31
HallC32×C5 ⋊ D217
HallC5×C7 ⋊ C3⋊D93
SylowC31 ⋊ C5×C3⋊D21
SylowC32 ⋊ C5×D217
SylowC7 ⋊ C5×C3⋊D93

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D651
commutator, cosocle, hypocenterC32×C7×C31C2×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C31C2
LCSG ▷ C32×C7×C31
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 9765

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5)(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)(38,39)(40,41)(42,43)(45,46)(48,49);(6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8)(37,38,40,42,43,41,39)(47,48,49);(44,45,46)

Conjugacy classes & character table

4890 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 198);