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×#62931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7074 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)(14,20)(18,22)(21,26)(25,27);(4,5,8,7)(6,10,11,9)(12,13,17,16)(14,18,20,22)(15,19,23,24)(21,25,26,27);(4,6,8,11)(5,9,7,10)(12,14)(13,18)(15,21)(16,22)(17,20)(19,25)(23,26)(24,27);(4,7,8,5)(6,10,11,9)(12,15)(13,19)(14,21)(16,24)(17,23)(18,25)(20,26)(22,27);(12,16,17,13)(14,22,20,18)(15,19,23,24)(21,25,26,27);(12,14)(13,18)(15,21)(16,22)(17,20)(19,25)(23,26)(24,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55919);