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C2×C3⋊(C3⋊(D71⋊C7)) #31822066

order17892
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊(C3⋊(D71⋊C7))
unambiguous semidirect product(C2×C3)⋊(C3⋊(D71⋊C7))
unambiguous semidirect product(C2×C71⋊C7)⋊(C3⋊D3)
unambiguous semidirect product(C2×C3⋊D213)⋊C7

Splittings

Fitting, socleC2×C32×C71 ⋊ C2×C7
derived, HallC32×C71 ⋊ C22×C7
HallC32×C71⋊C7 ⋊ C22
HallC2×C3⋊D213 ⋊ C7
HallC71⋊C7 ⋊ C2×C3⋊D3
SylowC71 ⋊ C2×C7×C3⋊D3
SylowC32 ⋊ C2×D71⋊C7

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C3⋊(D71⋊C7))
commutator, cosocle, hypocenterC32×C71C22×C7
FrattiniC1G
Fitting, socleC2×C32×C71C2×C7
LCSG ▷ C32×C71
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 1278

A minimal faithful permutation representation — 3 generators on 79 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17892, 83);