Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#21675 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,11)(8,9)(12,13)(14,16)(15,17)(18,19)(20,21,25,26,24,27,22,23);(5,9)(10,11)(12,14,15,18)(13,16,17,19)(20,22,24,25)(21,26,27,23);(4,6,8,7)(5,10,9,11)(13,17)(16,19)(20,22,24,25)(21,26,27,23);(4,7,8,6)(5,11,9,10)(20,23,22,27,24,26,25,21)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14663);