Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#16707 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,8)(6,10)(12,13,15,17)(14,18,19,16)(20,21,24,25)(22,26,27,23)(29,30)(31,35);(14,19)(16,18)(21,25)(22,27)(28,29,32,30)(31,33,35,34);(4,5,7,8)(6,9,10,11)(12,14,15,19)(13,16,17,18)(20,22,24,27)(21,23,25,26)(28,30,32,29)(31,33,35,34);(4,6)(5,9)(7,10)(8,11)(12,13)(14,16)(15,17)(18,19)(20,23)(21,22)(24,26)(25,27)(28,31)(29,33)(30,34)(32,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9695);