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×#17580 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
1024 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)(22,28,30,35)(23,27)(24,26)(25,29,33,31)(32,34);(4,6)(5,9)(7,13)(8,14)(10,17)(11,12)(15,18)(16,19)(20,22)(21,25)(23,31)(24,30)(26,33)(27,35)(28,34)(29,32);(21,26)(25,33)(27,34)(28,35);(4,7,8,16)(5,10,11,15)(6,13,14,19)(9,17,12,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10568);