Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#8730 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #381 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×#376 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,6,8,7,9,10,11)(12,13)(14,21,23,17)(15,20)(16,19)(18,22,25,27)(24,26);(12,14)(13,17)(15,22)(16,23)(18,20)(19,21)(24,27)(25,26);(12,14)(13,18,19,25)(15,22)(16,23)(17,20,21,26)(24,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1718);