Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#12980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C8×D4 |
| prime radical | C2×C3×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C16 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,7)(8,9,13,19,14,20,33,24,15,21,34,38,12,18,23,36)(10,22,17,37,25,11,27,29,26,28,31,39,30,16,35,32);(4,5,6,7)(8,10,23,27,12,30,34,17,15,26,33,35,14,25,13,31)(9,16,36,37,18,28,38,32,21,11,24,39,20,22,19,29);(8,11)(9,17)(10,24)(12,22)(13,32)(14,28)(15,16)(18,35)(19,30)(20,27)(21,31)(23,39)(25,38)(26,36)(29,34)(33,37)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5968);