Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
914 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)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,23,27,24,26,30,33)(22,25,29,32,28,31,34,35);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,25)(23,29)(24,28)(26,31)(27,32)(30,34)(33,35);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(21,26)(23,30)(25,31)(29,34);(5,11)(9,12)(10,19)(17,18)(22,28)(25,31)(29,34)(32,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14820);