Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(6,10)(8,11)(12,13,17,16)(14,18,22,21)(15,23,25,19)(20,27,24,26)(28,29)(30,35)(31,39)(32,37)(33,34)(36,41)(38,40)(42,43);(4,5,7,9)(6,8,10,11)(13,16)(18,21)(19,23)(26,27)(29,34)(30,37)(36,39)(38,43);(4,6)(5,8)(7,10)(9,11)(12,14,17,22)(13,18,16,21)(15,24,25,20)(19,27,23,26)(28,30,33,37)(29,35,34,32)(31,38,41,43)(36,42,39,40);(12,15,17,25)(13,19,16,23)(14,20,22,24)(18,26,21,27)(28,31,33,41)(29,36,34,39)(30,38,37,43)(32,40,35,42);(12,16,17,13)(14,21,22,18)(15,23,25,19)(20,27,24,26)(28,32,33,35)(29,37,34,30)(31,40,41,42)(36,43,39,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34391);