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×#62922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7072 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)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27);(12,13,16,19)(14,17,22,20)(15,18,23,25)(21,24,27,26);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,18)(16,19)(20,22)(21,24)(23,25)(26,27);(6,10)(9,11)(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(4,6,7,10)(5,9,8,11);(4,5)(6,9)(7,8)(10,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55910);