Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C73 · Schur multiplier (H2) = C73 |
| direct product | C7×#14558877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C72 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C73 |
| prime radical | C74 | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72 ◁ C73 ◁ G | |
| LECS | G ▷ C72 ▷ C7 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 343 |
A minimal faithful permutation representation — 3 generators on 56 points:
(1,2,3,4,5,6,7);(8,9,11,15,23,34,42,10,12,18,26,36,43,51,14,19,30,39,46,44,52,22,31,16,27,35,45,53,33,32,28,25,38,48,55,41,21,17,29,40,49,56,50,13,20,24,37,47,54);(9,12,19,31,32,21,13)(11,16,20,30,17,18,28)(15,24,29,25,27,39,26)(23,35,37,46,40,36,38)(34,43,44,45,48,49,47)679 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 69);