Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#37673 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(9,11)(12,13)(14,18)(15,17)(16,19);(5,7)(12,13,15,17)(14,16,19,18);(4,5)(6,7)(8,9,10,11);(12,14)(13,16)(15,19)(17,18);(4,5)(6,7)(12,13)(14,16)(15,17)(18,19)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30661);