Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#33071 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,7,8)(6,9,10,11)(12,13)(14,16)(15,17)(18,19)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,35)(33,34);(5,8)(9,11)(12,14,15,18)(13,16,17,19);(4,6,7,10)(5,9,8,11)(13,17)(16,19)(21,25)(26,34)(27,32)(28,35);(20,22,23,29)(21,26,25,34)(24,33,31,30)(27,35,32,28)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26059);