Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×Q8 ◁ G | |
2018 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)(8,9)(10,18,21,14)(11,15)(12,16)(13,17)(19,32,35,26)(20,33,37,27)(22,28)(23,29)(24,30)(25,31)(34,39,36,38);(4,5,6,7)(8,10)(9,14)(11,19)(12,20)(13,21)(15,26)(16,27)(17,18)(22,34)(23,35)(24,36)(25,37)(28,38)(29,32)(30,39)(31,33);(8,11,13,23)(9,15,17,29)(10,19,21,35)(12,24,25,22)(14,26,18,32)(16,30,31,28)(20,36,37,34)(27,39,33,38);(8,12,13,25)(9,16,17,31)(10,20,21,37)(11,22,23,24)(14,27,18,33)(15,28,29,30)(19,34,35,36)(26,38,32,39)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13343);