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×#17594 |
| 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 | |
1014 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)(7,15)(8,11)(10,16)(13,17,19,18)(20,21)(22,30)(23,29)(24,27)(25,28)(26,31)(32,35)(33,34);(4,6)(5,9)(7,13)(8,14)(10,17)(11,12)(15,18)(16,19)(20,22)(21,26)(23,31)(24,33)(25,32)(27,30)(28,34)(29,35);(20,23,25,24)(21,27,28,29)(22,31,32,33)(26,30,34,35);(4,7,8,16)(5,10,11,15)(6,13,14,19)(9,17,12,18)(21,28)(26,34)(27,29)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10582);