Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#8790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #382 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(5,10)(6,11)(7,8)(12,13)(14,22,25,18)(15,26,28,41)(16,21)(17,20)(19,27,32,42)(23,36)(24,35,39,31)(29,33)(30,37)(34,43)(38,40);(4,5,8,6,9,11,7,10)(12,14,16,24,17,25,29,39)(13,18,21,31,20,22,33,35)(15,23,27,37,28,38,42,43)(19,30,26,40,32,34,41,36);(4,6,7,5,9,10,8,11)(12,15)(13,19,20,32)(14,23)(16,27)(17,28)(18,30,22,34)(21,26,33,41)(24,37)(25,38)(29,42)(31,40,35,36)(39,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1778);