Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#37146 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(6,10)(8,11)(15,22)(19,24)(21,26)(25,27)(28,29,33,34)(30,31,36,39)(32,35,40,41)(37,38,42,43);(4,5,7,9)(6,8,10,11)(13,17)(14,20)(19,24)(21,26)(29,34)(30,36)(35,41)(37,42);(4,6,7,10)(5,8,9,11)(12,13)(14,18)(15,19)(16,17)(20,23)(21,25)(22,24)(26,27)(28,30)(29,31)(32,37)(33,36)(34,39)(35,38)(40,42)(41,43);(12,14,16,20)(13,18,17,23)(15,21,22,26)(19,25,24,27)(28,31)(29,30)(32,38)(33,39)(34,36)(35,37)(40,43)(41,42);(12,15)(13,19)(14,21)(16,22)(17,24)(18,25)(20,26)(23,27)(28,32,33,40)(29,35,34,41)(30,37,36,42)(31,38,39,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30134);