Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#41414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,13,14)(10,11,16,19)(12,15,20,21)(17,18,22,23)(24,25)(26,33)(27,28)(29,30)(31,36)(32,38)(34,35)(37,39);(9,14)(10,16)(15,21)(17,22)(25,30)(28,32)(31,33)(35,39);(4,5,6,7)(8,10)(9,11)(12,17)(13,16)(14,19)(15,18)(20,22)(21,23)(24,26,29,36)(25,31,30,33)(27,37,38,34)(28,35,32,39);(8,11)(9,10)(12,18)(13,19)(14,16)(15,17)(20,23)(21,22)(24,27)(25,32)(26,34)(28,30)(29,38)(31,35)(33,39)(36,37);(8,12,13,20)(9,15,14,21)(10,17,16,22)(11,18,19,23)(24,28,29,32)(25,27,30,38)(26,35,36,39)(31,37,33,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34402);