Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13404 |
| 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 | |
406 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)(6,15)(7,9)(8,22)(12,28)(13,23)(14,20)(16,31)(17,24)(18,33)(19,32)(21,25)(26,30)(29,34);(4,5)(6,16)(7,20)(8,23)(9,11)(10,14)(12,18)(13,25)(15,30)(17,29)(19,26)(21,27)(22,35)(24,31)(28,32)(33,34);(4,6)(5,12,14,16)(7,19)(8,24)(9,17)(10,18)(11,26,20,29)(13,31,23,28)(15,22)(21,33)(25,32)(27,30,35,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6392);