Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20338 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C42 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C22×#95 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
2036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14,17,27)(7,11)(8,12)(9,13)(10,16,22,30)(15,26,28,34)(18,23)(19,24)(20,25)(21,29,32,35)(31,33);(4,6)(5,10,13,22)(7,15)(8,16)(9,17)(11,21,23,32)(12,14,25,27)(18,28)(19,29)(20,30)(24,26,33,34)(31,35);(4,7)(5,11)(6,15)(8,19)(9,18)(10,21)(12,24)(13,23)(14,26)(16,29)(17,28)(20,31)(22,32)(25,33)(27,34)(30,35);(7,18)(11,23)(15,28)(19,31)(21,32)(24,33)(26,34)(29,35)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13326);