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×#62820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7382 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)(15,21)(16,19)(17,18)(22,26)(23,27)(24,25);(6,10)(9,11)(13,18)(14,21)(16,24)(23,26);(4,5)(6,9)(7,8)(10,11)(12,14)(13,15)(16,22)(17,21)(18,20)(19,23)(24,27)(25,26);(12,15)(13,14)(16,23)(17,20)(18,21)(19,22)(24,26)(25,27);(4,6)(5,9)(7,10)(8,11)(12,16,17,24)(13,19,18,25)(14,22,21,27)(15,23,20,26);(4,5)(6,9)(7,8)(10,11)(12,15)(13,14)(16,23)(17,20)(18,21)(19,22)(24,26)(25,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55808);