Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C42 |
| direct product | C3×#16713 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,8)(6,10)(12,13,15,16)(14,18,19,17)(20,21,23,25)(22,26,27,24)(28,29)(30,35)(31,34)(32,33);(13,16)(17,18)(22,27)(24,26)(28,30,33,31)(29,34,32,35);(4,5,7,8)(6,9,10,11)(12,13)(14,18)(15,16)(17,19)(20,22,23,27)(21,24,25,26)(28,31,33,30)(29,34,32,35);(4,6)(5,9)(7,10)(8,11)(12,14,15,19)(13,17,16,18)(20,21)(22,24)(23,25)(26,27)(28,32,33,29)(30,35,31,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9701);