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C7×C3⋊D465 #34086291

order19530
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊D465
unambiguous semidirect productC3⋊(C7×D465)
unambiguous semidirect productC5⋊(C7×C3⋊D93)
unambiguous semidirect productC31⋊(C7×C3⋊D15)

Splittings

Fitting, socle, HallC32×C5×C7×C31 ⋊ C2
derived, HallC32×C5×C31 ⋊ C2×C7
HallC32×C7×C31 ⋊ D5
HallC5×C7×C31 ⋊ C3⋊D3
HallC32×C5×C7 ⋊ D31
HallC32×C31 ⋊ C7×D5
HallC7×C31 ⋊ C3⋊D15
HallC5×C31 ⋊ C7×C3⋊D3
HallC32×C7 ⋊ D155
HallC32×C5 ⋊ C7×D31
HallC5×C7 ⋊ C3⋊D93
SylowC31 ⋊ C7×C3⋊D15
SylowC32 ⋊ C7×D155
SylowC5 ⋊ C7×C3⋊D93

Subgroups

subgroupG/H
center, hypercenterC7C3⋊D465
commutator, cosocle, hypocenterC32×C5×C31C2×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C31C2
LCSG ▷ C32×C5×C31
UCSC1 ◁ C7

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:

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

Conjugacy classes & character table

4893 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 196);