Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21075 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#121 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1014 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)(4,5,7,6)(8,9)(10,19)(11,24)(12,17)(13,28)(14,18)(15,22)(16,27)(20,34)(21,30)(23,33)(25,39)(26,31)(29,35)(32,37)(36,38);(4,6,7,5)(8,10,13,22,14,23,12,21)(9,15,17,19,18,30,28,33)(11,20,26,36,27,37,25,35)(16,29,31,32,24,38,39,34);(8,11,13,26,14,27,12,25)(9,16,17,31,18,24,28,39)(10,20,22,36,23,37,21,35)(15,29,19,32,30,38,33,34);(8,12,14,13)(9,17,18,28)(10,21,23,22)(11,25,27,26)(15,19,30,33)(16,31,24,39)(20,35,37,36)(29,32,38,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14063);