Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#8158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C24×C3 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3)(4,5)(6,9)(7,11)(8,10)(12,16)(13,15)(14,17)(18,19)(20,21,24,28,25,29,36,42)(22,30,33,44,32,26,37,39)(23,27,34,40,35,41,47,50)(31,43,46,51,45,38,48,49);(4,6,7,12,8,13,14,18)(5,9,11,16,10,15,17,19)(20,22)(21,26,29,30)(23,31)(24,33)(25,32)(27,38,41,43)(28,39,42,44)(34,46)(35,45)(36,37)(40,49,50,51)(47,48);(5,10)(9,15)(11,17)(16,19)(20,23,24,34,25,35,36,47)(21,27,28,40,29,41,42,50)(22,31,33,46,32,45,37,48)(26,38,39,49,30,43,44,51)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1146);