Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17571 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C3×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,12,14,9)(7,11)(8,10)(13,18,19,16)(15,17)(20,21)(22,29,31,34)(23,32)(24,27)(25,28)(26,30,35,33);(4,6)(5,9)(7,13)(8,14)(10,12)(11,16)(15,19)(17,18)(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(5,10)(9,12)(11,17)(16,18)(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29);(4,7,8,15)(5,11,10,17)(6,13,14,19)(9,16,12,18)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10559);