Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#20776 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#117 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#383 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×M16 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,15)(7,19,21,26)(8,13)(9,14)(10,18)(11,20,27,33)(12,22)(16,23,29,34)(17,25)(24,31)(28,30,32,35);(4,6,9,17)(5,10)(7,16,21,29)(8,18,22,31)(11,23)(12,15)(13,25)(14,24)(19,32)(20,30,33,35)(26,28)(27,34);(4,7,8,20,9,21,22,33)(5,11,12,26,14,27,13,19)(6,16,18,30,17,29,31,35)(10,23,15,28,24,34,25,32);(6,17)(10,24)(15,25)(16,29)(18,31)(23,34)(28,32)(30,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13764);