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×#62905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7326 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)(5,7)(9,10)(12,13)(14,19)(15,17)(16,23)(18,22)(20,24)(21,25)(26,27);(4,5,8,7)(6,10,11,9)(12,14)(13,18)(15,19)(16,21)(17,22)(20,25)(23,26)(24,27);(4,6,8,11)(5,9,7,10)(12,13,17,15)(14,19,22,18)(16,20,24,23)(21,26,27,25);(4,7,8,5)(6,10,11,9)(12,15,17,13)(14,19,22,18)(16,23,24,20)(21,26,27,25);(12,14)(13,19)(15,18)(16,21)(17,22)(20,26)(23,25)(24,27);(12,16)(13,20)(14,21)(15,23)(17,24)(18,25)(19,26)(22,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55893);