Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #157⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#157⋊D3 |
| unambiguous semidirect product | #157⋊(D3×C7) |
| Fitting, socle, Hall | C33×C7 ⋊ D4 |
| Sylow | C33 ⋊ C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #157⋊D3 |
| commutator, cosocle | C3×C3⋊D3 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C33×C7 | D4 |
| hypocenter | C33 | C7×D4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C7 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5,6,7)(8,9,10)(11,15,13,12)(14,16);(9,10)(14,15)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 826);