Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C49 · Schur multiplier (H2) = C72 |
| unambiguous semidirect product | H(3,𝔽7)⋊C49 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C72 | H(3,𝔽7) |
| commutator | C72 | C7×C49 |
| Frattini, cosocle | C73 | C72 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72 ◁ C73 ◁ G | |
| LECS | G ▷ C73 ▷ C7 ▷ C1 | |
103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 98 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 2 generators on 98 points:
(1,2,4,7,11,16,22,3,5,8,12,17,23,29,6,9,13,18,24,30,35,10,14,19,25,31,36,40,15,20,26,32,37,41,44,21,27,33,38,42,45,47,28,34,39,43,46,48,49)(50,51,53,58,68,80,88)(52,56,61,71,82,90,93)(54,62,69,83,92,57,66)(55,64,73,84,94,89,98)(59,72,85,95,87,65,60)(63,74,81,91,97,67,78)(70,75,86,96,79,77,76);(50,51,54,58,69,83,93)(52,56,63,71,81,91,92)(53,59,68,80,89,87,65)(55,61,73,82,90,88,98)(57,66,76,62,75,86,97)(60,74,72,85,96,67,78)(64,70,84,94,95,79,77)Polycyclic presentation:
7^5|a^7=d,b^a=bc,c^a=ce,c^b=ce679 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 24);