Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#17365 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
1214 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)(4,5,7,11,8,9,6,10)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25);(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(5,9)(10,11);(4,6,8,7)(5,10,9,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10353);