Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,6)(8,9)(10,17)(11,21)(12,15)(13,16)(14,18)(19,22)(20,23);(4,5,7,6)(8,9)(10,17)(11,15)(12,21)(13,16)(14,19)(18,22)(20,23);(8,10,13,20)(9,14,16,22)(11,19,12,18)(15,23,21,17);(4,6,7,5)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17220);