Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#7633 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C2×C3×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,16)(15,17)(18,19)(20,21,23,25,24,26,22,27);(4,6)(5,8)(7,11)(9,10)(13,16)(14,19)(17,18);(5,9)(8,10)(13,17)(16,18)(20,22,24,23)(21,25,26,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 621);