Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#9875 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,17,18,16,19,15,14)(20,21)(22,28)(23,32)(24,26)(25,30)(27,29)(31,34)(33,35);(5,8)(9,10)(12,14,15,19,16,18,17,13)(20,22,29,35,24,31,23,30)(21,25,32,34,26,33,27,28);(4,6,7,11)(5,9,8,10)(12,15,16,17)(13,18,19,14)(21,26)(25,33)(27,32)(28,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2863);