Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C74 · Schur multiplier (H2) = C77 |
| direct product | C72×H(3,𝔽7) |
| unambiguous semidirect product | C72⋊C73 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C73 | C72 |
| commutator, Frattini, cosocle | C7 | C74 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C73 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
7678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 4 generators on 63 points:
(1,2,3,4,5,6,7);(8,9,10,11,12,13,14);(15,16,18,22,29,40,49)(17,20,24,31,42,51,58)(19,25,35,47,56,63,57)(21,27,34,44,53,50,59)(23,32,45,55,62,48,26)(28,38,33,46,41,52,60)(30,43,54,61,39,37,36);(16,19,26,37,38,27,20)(18,23,33,24,25,36,34)(22,30,31,32,44,35,46)(29,41,47,53,45,42,43)(40,50,54,56,51,52,55)(49,58,59,60,61,62,63)2695 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 77);