Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62921 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(5,8)(9,11)(12,13,17,18)(14,15,20,23)(16,19,24,25)(21,22,26,27);(13,18)(14,20)(19,25)(21,26);(4,5)(6,9)(7,8)(10,11)(12,14)(13,15)(16,21)(17,20)(18,23)(19,22)(24,26)(25,27);(6,10)(9,11)(12,15)(13,14)(16,22)(17,23)(18,20)(19,21)(24,27)(25,26);(4,6,7,10)(5,9,8,11);(4,5)(6,9)(7,8)(10,11)(12,16,17,24)(13,19,18,25)(14,21,20,26)(15,22,23,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55909);