Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#33404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#106 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
636 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)(5,10,12,21)(7,17)(8,19)(11,20,23,31)(13,26,25,22)(15,29)(16,28)(18,30)(24,34,33,32)(27,35);(4,5,7,10,8,13,17,22,9,12,18,21,19,25,30,26)(6,11,15,20,16,24,29,32,14,23,27,31,28,33,35,34);(4,6)(5,11)(7,15)(8,16)(9,14)(10,20)(12,23)(13,24)(17,29)(18,27)(19,28)(21,31)(22,32)(25,33)(26,34)(30,35);(6,14)(11,23)(15,27)(16,28)(20,31)(24,33)(29,35)(32,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26392);