Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17960 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,20,17,14,21,19)(15,22,23,26,24,18,25,27);(5,8)(6,11)(9,10)(12,14)(13,17)(15,22)(16,19)(18,24)(20,21)(23,26)(25,27);(5,9)(8,10)(12,15,16,23,17,24,21,25)(13,18,20,27,14,22,19,26);(13,14)(18,22)(19,20)(26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10948);