Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
494 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)(4,5,8,11,9,13,7,12)(6,14,16,18,17,19,15,10)(20,21)(22,30,23,26)(24,29)(25,28)(27,32,34,31)(33,35);(4,6)(5,10,13,18)(7,15)(8,16)(9,17)(11,14,12,19)(20,22,24,23)(21,26)(25,32,33,31)(27,28)(29,30)(34,35);(5,11)(7,8)(10,14)(12,13)(15,16)(18,19)(20,23,24,22)(21,27,29,34)(25,31,33,32)(26,28,30,35)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2165);