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×#62917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7186 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)(12,13,19,22)(14,23,17,20)(15,18,16,21)(24,25,26,27);(12,14,19,17)(13,20,22,23)(15,26,16,24)(18,25,21,27);(4,5)(6,9)(7,8)(10,11)(12,15,19,16)(13,21,22,18)(14,24,17,26)(20,25,23,27);(6,10)(9,11)(12,16,19,15)(13,18,22,21)(14,24,17,26)(20,25,23,27);(4,6,7,10)(5,9,8,11)(12,17,19,14)(13,20,22,23)(15,26,16,24)(18,27,21,25);(4,5)(6,9)(7,8)(10,11)(12,18)(13,15)(14,25)(16,22)(17,27)(19,21)(20,26)(23,24)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55905);