Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30441 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
902 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)(6,16)(7,13)(8,11)(9,14)(10,15)(12,17)(18,32)(19,24)(20,29)(21,25)(22,28)(23,27)(26,33)(30,34)(31,35);(5,11)(6,17)(7,20)(8,9)(13,28)(14,15)(16,24)(19,23)(21,22)(26,32)(30,35)(31,33);(4,6)(5,12)(7,18)(8,17)(9,23)(10,19)(11,24)(13,26)(14,16)(15,27)(20,33)(21,31)(22,34)(25,32)(28,35)(29,30);(4,7)(5,13)(6,18)(8,21)(9,20)(10,22)(11,25)(12,26)(14,29)(15,28)(16,30)(17,31)(19,34)(23,33)(24,32)(27,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23429);