Finite Group DatabasebetaFinite Group Databasebeta| order | 351 |
| isoclinism | isoclinic class of C33⋊C13 |
| automorphism | ? |
| homology | Abelianization (H1) = C13 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C33⋊C13 |
| Fitting, socle, derived, Sylow | C33 ⋊ C13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C33 | C13 |
| LCS | G ▷ C33 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,4,11,24,20,27,21,9,18,6,14,16,15)(3,8,19,7,17,13,5,12,25,10,22,26,23);(1,18,8)(2,14,4)(3,25,20)(5,12,7)(6,26,10)(9,22,11)(13,23,16)(15,17,21)(19,24,27)Polycyclic presentation:
13,3^3|b^a=d,c^a=b^2d,d^a=b^2c^2d15 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(351, 12);