Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C8 · Schur multiplier (H2) = C3 |
| direct product | C7×(C3⋊(C3⋊C8))⋊C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C8))⋊(C3×C7) |
| derived | C32 ⋊ C3×C7×C8 |
| Hall | C7×H(3,𝔽3) ⋊ C8 |
| Sylow | H(3,𝔽3) ⋊ C7×C8 |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(17,22,24,19,20,23)(18,21);(16,17,19)(18,20,23)(21,24,22)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 69);