Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#9848 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C23×C3 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,11,9,12,7,17)(6,13,16,18,14,19,15,10)(20,21)(22,28)(23,25)(24,32)(26,31)(27,29)(30,34)(33,35);(4,6)(5,10,12,18)(7,15)(8,16)(9,14)(11,13,17,19)(21,25)(26,33)(27,32)(28,34);(6,14)(10,18)(13,19)(15,16)(20,22,29,35,23,30,24,31)(21,26,32,34,25,33,27,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2836);