Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#7957 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #95 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C2×#95 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(6,7)(10,11)(12,13)(14,22,25,26)(15,19)(16,20)(17,21)(18,24,27,23);(4,5,7,10,8,11,6,9)(12,14)(13,18,21,27)(15,23)(16,24)(17,25)(19,26,20,22);(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,27,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 945);