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C2×C3⋊(C5⋊(D71⋊C7)) #44447199

order29820
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊(C5⋊(D71⋊C7))
unambiguous semidirect product(C2×C3)⋊(C5⋊(D71⋊C7))
unambiguous semidirect product(C2×C5)⋊(C3⋊(D71⋊C7))
unambiguous semidirect product(C2×C71⋊C7)⋊D15
unambiguous semidirect product(C2×D1065)⋊C7

Splittings

Fitting, socleC2×C3×C5×C71 ⋊ C2×C7
derived, HallC3×C5×C71 ⋊ C22×C7
HallC3×C5×C71⋊C7 ⋊ C22
HallC2×D1065 ⋊ C7
HallC5×C71⋊C7 ⋊ C2×D3
HallC3×C71⋊C7 ⋊ C2×D5
HallC71⋊C7 ⋊ C2×D15
HallC5×C71 ⋊ C2×D3×C7
HallC3×C71 ⋊ C2×C7×D5
SylowC71 ⋊ C2×C7×D15
HallC3×C5 ⋊ C2×D71⋊C7
SylowC5 ⋊ C2×C3⋊(D71⋊C7)
SylowC3 ⋊ C2×C5⋊(D71⋊C7)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C5⋊(D71⋊C7))
commutator, cosocle, hypocenterC3×C5×C71C22×C7
FrattiniC1G
Fitting, socleC2×C3×C5×C71C2×C7
LCSG ▷ C3×C5×C71
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 2130

A minimal faithful permutation representation — 2 generators on 81 points:

(4,5)(7,8)(9,10)(12,16,17,34,30,64,65,13,14,22,24,47,45,73)(15,29,21,49,50,18,41,19,20,31,32,72,26,27)(23,56,53,62,58,46,59,36,37,75,42,38,67,68)(25,51,60,78,70,71,33,39,40,74,57,48,66,52)(28,63,35,76,77,80,61,43,44,54,55,69,81,79);(1,2)(3,4,5)(6,7,9,10,8)(11,12,15,21,33,53,40,49,37,57,79,27,42,55,74,63,80,77,67,20,30,48,71,64,43,26,16,23,38,59,65,72,24,39,22,35,54,17,25,34,50,73,68,58,36,14,18,28,45,66,70,47,29,46,69,81,44,60,76,62,41,61,78,56,32,51,75,52,31,19,13)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 87);