Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#11059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
842 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)(8,9,13,19,14,17,12,18)(10,15,21,31,22,30,28,35)(11,23,26,33,27,16,25,34)(20,36,24,39,37,29,38,32);(4,5,6,7)(8,10)(9,15)(11,24)(12,28)(13,21)(14,22)(16,32)(17,30)(18,35)(19,31)(20,25)(23,39)(26,37)(27,38)(29,33)(34,36);(8,11,14,27)(9,16,17,23)(10,20,22,37)(12,26,13,25)(15,29,30,36)(18,34,19,33)(21,38,28,24)(31,39,35,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4047);