Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33380 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9)(10,19)(11,17)(12,24)(13,28)(14,16)(15,21)(18,26)(20,36)(22,35)(23,33)(25,31)(27,38)(29,34)(30,37)(32,39);(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,30)(17,31)(18,34)(19,24)(21,26)(23,37)(27,39)(28,32)(33,36)(35,38);(9,16)(15,30)(17,28)(18,33)(19,35)(24,38)(31,32)(34,36);(8,11,14,13)(9,17,16,28)(10,20,22,29)(12,23,27,26)(15,31,30,32)(18,24,33,38)(19,36,35,34)(21,25,37,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26368);