Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1869 |
| prime radical | C2×C3×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C8 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,10,15,9,14,28,32,11,16,30,34,8,27,18,19)(6,17,21,35,22,7,24,26,23,25,13,33,20,12,31,29);(4,6,18,31,8,20,30,13,11,23,28,24,9,22,10,21)(5,12,32,35,27,25,15,29,16,7,19,33,14,17,34,26);(4,7)(5,13)(6,19)(8,17)(9,25)(10,29)(11,12)(14,31)(15,22)(16,21)(18,33)(20,34)(23,32)(24,27)(26,30)(28,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5974);