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D7×C3⋊(D71⋊C5) #44447187

order29820
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productD7×C3⋊(D71⋊C5)
unambiguous semidirect product(C3×D7)⋊(D71⋊C5)
unambiguous semidirect product(D7×C71⋊C5)⋊D3
unambiguous semidirect product(D7×D213)⋊C5

Splittings

Fitting, socle, derived, HallC3×C7×C71 ⋊ C22×C5
HallC3×C7×C71⋊C5 ⋊ C22
HallD7×D213 ⋊ C5
HallC7×C71⋊C5 ⋊ C2×D3
HallC3×C71⋊C5 ⋊ C2×D7
HallC7×C71 ⋊ C2×C5×D3
HallC71⋊C5 ⋊ D3×D7
HallC3×C71 ⋊ C2×C5×D7
SylowC71 ⋊ C5×D3×D7
HallC3×C7 ⋊ C2×D71⋊C5
SylowC7 ⋊ C2×C3⋊(D71⋊C5)
SylowC3 ⋊ D7×D71⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C71C22×C5
LCSG ▷ C3×C7×C71
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 1491

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

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 75);