Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21880 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(12,13,16,19,15,18,23,26)(14,20,22,27,21,17,24,25);(4,5)(6,9)(7,8)(10,11)(12,14)(13,17)(15,21)(16,22)(18,20)(19,25)(23,24)(26,27);(5,8)(9,11)(13,18)(14,21)(16,23)(25,27);(4,6,7,10)(5,9,8,11)(13,18)(17,20)(19,26)(25,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14868);