Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C23 |
| direct product | C3×#7258 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #102 |
| commutator | C2×C8 | C3×C42 |
| Frattini, cosocle | C23×C8 | C22×C3 |
| prime radical | C24×C3 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3)(4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17)(18,21)(19,28)(20,24)(22,25)(23,26)(27,30)(29,31);(9,12,14,13)(17,21)(18,25)(19,26)(22,23)(24,30)(27,31)(28,29)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 246);