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×#62961 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7302 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)(14,21)(15,24)(16,19)(17,23)(18,20)(22,26)(25,27);(4,5,8,7)(6,10,11,9)(12,14)(13,17)(15,25)(16,22)(18,23)(19,27)(20,21)(24,26);(4,6,8,11)(5,9,7,10)(12,15,18,16)(13,19,20,24)(14,22,23,25)(17,26,21,27);(4,7,8,5)(6,10,11,9)(12,14)(13,17)(15,22)(16,25)(18,23)(19,26)(20,21)(24,27);(12,16,18,15)(13,19,20,24)(14,22,23,25)(17,27,21,26);(12,17)(13,14)(15,26)(16,27)(18,21)(19,22)(20,23)(24,25)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55949);