Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C42 |
| direct product | C3×#8138 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C42 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #376 |
| prime radical | C23×C3 | #117 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,7,9,6,8,10,11)(12,13)(14,22,25,35)(15,19)(16,20)(17,21)(18,24,30,38)(23,34,37,42)(26,31)(27,32)(28,33)(29,36,40,43)(39,41);(5,8)(9,11)(12,14)(13,18,21,30)(15,23)(16,24)(17,25)(19,29,32,40)(20,22,33,35)(26,36)(27,37)(28,38)(31,34,41,42)(39,43);(12,15,16,26,17,27,28,39)(13,19,20,31,21,32,33,41)(14,23,24,36,25,37,38,43)(18,29,22,34,30,40,35,42)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1126);