Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62819 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(5,8)(9,11)(12,13)(14,20)(15,17)(16,23)(18,22)(19,24)(21,25)(26,27);(6,10)(9,11)(12,14,17,22)(13,18,15,20)(16,21,24,27)(19,25,23,26);(4,5)(6,9)(7,8)(10,11)(12,15,17,13)(14,18,22,20)(16,23,24,19)(21,25,27,26);(12,16)(13,19)(14,21)(15,23)(17,24)(18,25)(20,26)(22,27);(4,6)(5,9)(7,10)(8,11)(12,14)(13,20)(15,18)(16,21)(17,22)(19,26)(23,25)(24,27);(4,5)(6,9)(7,8)(10,11)(12,16)(13,19)(14,21)(15,23)(17,24)(18,25)(20,26)(22,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55807);