Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,11)(6,16)(7,9)(8,22)(12,27)(13,30)(14,15)(17,35)(18,23)(19,33)(20,34)(21,24)(25,28)(29,31);(4,5)(6,17)(7,14)(8,13)(9,11)(10,15)(12,18)(16,31)(19,29)(20,25)(21,32)(22,26)(23,35)(24,30)(27,34)(28,33);(4,6)(5,12,15,29)(7,19)(8,23)(9,18)(10,20)(11,25,14,17)(13,31,30,27)(16,22)(21,34)(24,33)(26,28,32,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6390);