Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17963 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,13)(7,12)(8,16)(9,11)(10,14)(15,19)(17,18)(20,21,25,28,26,22,24,29)(23,30,32,35,33,27,31,34);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,22)(21,26)(23,30)(24,28)(25,29)(27,33)(31,34)(32,35);(5,11)(10,18)(12,16)(13,19)(20,23,25,32,26,33,24,31)(21,27,28,34,22,30,29,35);(4,7,9,8)(5,12,11,16)(6,14,15,17)(10,19,18,13)(20,24,26,25)(21,28,22,29)(23,31,33,32)(27,34,30,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10951);