Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17593 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
1042 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)(5,11,13,25)(6,16)(7,21)(8,22)(12,27,28,35)(14,26,15,23)(18,32)(19,33)(20,31)(24,29,34,30);(4,5)(6,12)(7,11)(8,23)(9,14)(10,15)(13,17)(16,30)(18,24)(19,27)(20,29)(21,26)(22,25)(28,31)(32,34)(33,35);(4,6,17,31)(5,12,13,28)(7,18,22,32)(8,19,21,33)(9,20,10,16)(11,24,25,34)(14,29,15,30)(23,27,26,35);(4,7,10,8)(5,11,15,23)(6,18,16,19)(9,21,17,22)(12,24,30,27)(13,25,14,26)(20,33,31,32)(28,34,29,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10581);