Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20360 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C22⋊C4 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C22×#95 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×#41 ◁ G | |
1996 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)(6,14,16,10)(7,11)(8,13)(9,12)(15,26,28,21)(17,27,30,22)(18,23)(19,24)(20,25)(29,34,35,32)(31,33);(4,6,8,17,9,16,20,30)(5,10,13,22,12,14,25,27)(7,15,19,29,18,28,31,35)(11,21,24,32,23,26,33,34);(4,7)(5,11)(6,15)(8,19)(9,18)(10,21)(12,23)(13,24)(14,26)(16,28)(17,29)(20,31)(22,32)(25,33)(27,34)(30,35);(7,18)(11,23)(15,28)(19,31)(21,26)(24,33)(29,35)(32,34)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13348);