Finite Group DatabasebetaFinite Group Databasebeta| order | 343 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C72 · Schur multiplier (H2) = C72 |
| extraspecial | 7+1+2 |
| unitriangular | UT(3,𝔽7) |
| Heisenberg | H(3,𝔽7) |
| unambiguous semidirect product | C72⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C72 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 49 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,4,8,15,26,35)(3,6,10,17,28,37,44)(5,11,21,33,42,49,43)(7,13,20,30,39,36,45)(9,18,31,41,48,34,12)(14,24,19,32,27,38,46)(16,29,40,47,25,23,22);(2,5,12,23,24,13,6)(4,9,19,10,11,22,20)(8,16,17,18,30,21,32)(15,27,33,39,31,28,29)(26,36,40,42,37,38,41)(35,44,45,46,47,48,49)Polycyclic presentation:
7^3|b^a=bc55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(343, 3);