Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13391 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2033 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,17)(7,21)(8,23)(9,25)(10,14)(11,16)(12,18)(13,22)(15,24)(19,29)(20,30)(26,28)(27,31)(32,35)(33,34);(4,6)(5,12)(7,8)(9,19)(10,18)(11,20)(13,31)(14,30)(15,29)(16,21)(17,23)(22,24)(25,33)(26,34)(27,35)(28,32);(4,7,8,6,11,18,10,20)(5,13,14,32,16,33,23,29)(9,19,24,34,27,35,26,22)(12,28,17,31,21,15,30,25)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6379);