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×#37147 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1388 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)(15,22)(19,24)(21,26)(25,27)(28,29)(30,34)(31,33)(32,39)(35,40)(36,38)(37,42)(41,43);(4,5,7,9)(6,8,10,11)(13,17)(14,20)(19,24)(21,26)(29,31)(34,36)(35,39)(41,42);(4,6,7,10)(5,8,9,11)(12,13)(14,18)(15,19)(16,17)(20,23)(21,25)(22,24)(26,27)(28,30,33,38)(29,34,31,36)(32,37,40,43)(35,41,39,42);(12,14,16,20)(13,18,17,23)(15,21,22,26)(19,25,24,27)(28,31,33,29)(30,36,38,34)(32,39,40,35)(37,42,43,41);(12,15)(13,19)(14,21)(16,22)(17,24)(18,25)(20,26)(23,27)(28,32,33,40)(29,35,31,39)(30,37,38,43)(34,41,36,42)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30135);