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×#62919 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7082 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)(14,22)(15,16)(17,19)(18,21)(20,27)(23,24)(25,26);(12,14,18,17)(13,19,21,22)(15,25,20,23)(16,24,27,26);(4,5)(6,9)(7,8)(10,11)(12,15,18,20)(13,16,21,27)(14,23,17,25)(19,26,22,24);(6,10)(9,11)(12,16)(13,20)(14,24)(15,21)(17,26)(18,27)(19,23)(22,25);(4,6,7,10)(5,9,8,11)(12,17,18,14)(13,19,21,22)(15,25,20,23)(16,26,27,24);(4,5)(6,9)(7,8)(10,11)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55907);