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×#62918 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7386 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)(14,19)(15,25)(16,26)(17,20)(18,24)(21,27)(22,23);(12,14)(13,18)(15,22)(16,23)(17,24)(19,20)(21,26)(25,27);(4,5)(6,9)(7,8)(10,11)(12,15)(13,16)(14,21)(17,26)(18,27)(19,23)(20,25)(22,24);(6,10)(9,11)(12,15)(13,16)(14,22)(17,26)(18,23)(19,27)(20,25)(21,24);(4,6,7,10)(5,9,8,11)(12,14)(13,19)(15,21)(16,23)(17,24)(18,20)(22,26)(25,27);(4,5)(6,9)(7,8)(10,11)(12,16)(13,15)(14,23)(17,25)(18,22)(19,21)(20,26)(24,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55906);