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) = C1 |
| extraspecial | 7−1+2 |
| unambiguous semidirect product | C49⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C72 |
| prime radical | C72 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 49 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,4,8,15,26,35,3,6,10,17,28,37,44,7,13,20,30,39,36,45,14,24,19,32,27,38,46,25,23,22,16,29,40,47,34,12,9,18,31,41,48,43,5,11,21,33,42,49);(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|a^7=c,b^a=bc55 conjugacy classes — view the full character table.
| Fitting subgroup of | D49⋊C7, (C49⋊C3)⋊C7 |
| maximal subgroup of | D49⋊C7, C2×7−1+2, C3×7−1+2, … |
| maximal normal subgroup of | D49⋊C7, C2×7−1+2, C3×7−1+2, … |
| maximal quotient of | C2×7−1+2, C3×7−1+2, C5×7−1+2, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(343, 4);