Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60469 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C2×C3 | C22×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
2628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,16)(7,21)(8,13)(9,14)(10,15)(11,19)(12,22)(17,31)(18,27)(20,26)(23,28)(24,29)(25,32)(30,33)(34,35);(4,6,8,19,10,18,9,20)(5,11,14,16,15,26,13,27)(7,17,22,33,23,32,24,34)(12,25,21,30,29,31,28,35);(4,7)(5,12)(6,17)(8,22)(9,24)(10,23)(11,25)(13,28)(14,21)(15,29)(16,30)(18,32)(19,33)(20,34)(26,31)(27,35);(6,18)(11,26)(16,27)(17,32)(19,20)(25,31)(30,35)(33,34);(4,8,10,9)(5,13,15,14)(6,19,18,20)(7,22,23,24)(11,27,26,16)(12,28,29,21)(17,33,32,34)(25,35,31,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53457);