Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60428 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
2866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(5,7)(8,9)(10,14)(11,13)(12,16)(15,19)(17,21)(18,20)(22,23);(4,5,6,7);(9,13)(11,19)(14,20)(15,16)(18,23)(21,22);(8,10)(9,14)(11,18)(12,17)(13,20)(15,22)(16,21)(19,23);(10,17)(14,21)(18,23)(20,22)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53416);