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D21⋊(D71⋊C5) #44447189

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

Aliases

unambiguous semidirect productD21⋊(D71⋊C5)
unambiguous semidirect product(C3⋊(D71⋊C5))⋊D7
unambiguous semidirect product(C7⋊(D71⋊C5))⋊D3
unambiguous semidirect product(D21⋊D71)⋊C5

Splittings

Fitting, socle, derived, HallC3×C7×C71 ⋊ C22×C5
HallC3×C7×C71⋊C5 ⋊ C22
HallD21⋊D71 ⋊ 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 ⋊ D3×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)(6,7)(8,9)(11,12,18,44,57)(13,23,34,38,67)(14,25,45,35,56)(15,31,76,54,39)(16,36,72,73,66)(17,40,64,53,77)(19,33,68,63,37)(20,49,30,71,32)(21,55,79,52,70)(22,58,59,27,65)(24,62,28,50,78)(26,60,48,41,46)(29,69,43,42,80)(47,81,75,74,51);(1,2,3)(5,6)(7,8)(9,10)(12,13)(14,16)(15,19)(17,22)(18,26)(20,29)(21,32)(23,37)(24,35)(25,38)(27,42)(28,45)(30,47)(31,50)(33,53)(34,56)(36,59)(39,57)(40,52)(41,46)(43,64)(44,55)(48,68)(49,72)(51,75)(54,71)(58,76)(60,74)(61,77)(62,66)(63,65)(67,70)(69,73)(78,81)(79,80)

Polycyclic presentation:

2^2,5,3,7,71|e^a=e^6,f^a=f^70,d^b=d^2,e^b=e^6,f^c=f^57

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 77);