Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#11046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C25×C3 ◁ G | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,9)(10,13)(11,18,19,14)(12,15)(16,20,23,22)(17,21);(4,5,6,7)(8,10)(9,13)(11,16)(12,17)(14,22)(15,21)(18,20)(19,23);(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,23)(21,22)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4034);