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×#17596 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
994 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)(6,13,16,10)(7,11)(8,17)(9,12)(14,19,15,18)(20,21)(22,28)(23,31)(24,26)(25,29)(27,33)(30,35)(32,34);(4,6,9,16)(5,10,12,13)(7,14,8,15)(11,18,17,19)(20,22,24,30)(21,25,26,34)(23,32,33,29)(27,35,31,28);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,18,13,19)(21,26)(25,34)(27,31)(28,35);(4,8,9,7)(5,11,12,17)(6,15,16,14)(10,18,13,19)(21,26)(25,34)(27,31)(28,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10584);