Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23434 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1916 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)(6,15)(7,20)(8,25)(9,14)(10,13)(11,17)(12,23)(16,33)(18,27)(19,28)(21,32)(22,30)(24,31)(26,34)(29,35);(4,6)(5,11)(7,21)(8,18)(9,17)(10,19)(12,29)(13,27)(14,28)(15,25)(16,22)(20,26)(23,34)(24,35)(30,32)(31,33);(4,7)(5,12)(6,16)(8,22)(9,23)(10,24)(11,26)(13,30)(14,31)(15,32)(17,21)(18,35)(19,34)(20,25)(27,33)(28,29);(5,13)(11,27)(12,30)(14,25)(15,28)(20,31)(26,33)(29,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16422);