Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#20954 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1694 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)(5,7)(8,9,13,19,14,17,12,18)(10,20,24,15,23,31,22,30)(11,25,27,32,28,16,26,33)(21,34,37,38,36,39,35,29);(4,5,6,7)(8,10)(9,15)(11,21)(12,24)(13,22)(14,23)(16,29)(17,30)(18,31)(19,20)(25,38)(26,37)(27,35)(28,36)(32,34)(33,39);(8,11,14,28)(9,16,17,25)(10,21,23,36)(12,27,13,26)(15,29,30,38)(18,33,19,32)(20,34,31,39)(22,37,24,35);(9,17)(15,30)(16,25)(18,19)(20,31)(29,38)(32,33)(34,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13942);