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×#62923 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7300 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)(12,13,17,20)(14,21,15,18)(16,24,25,19)(22,26,23,27);(12,14,17,15)(13,18,20,21)(16,22,25,23)(19,26,24,27);(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,18)(16,19)(17,20)(22,27)(23,26)(24,25);(6,10)(9,11)(12,13)(14,18)(15,21)(16,19)(17,20)(22,26)(23,27)(24,25);(4,6,7,10)(5,9,8,11)(12,15,17,14)(13,18,20,21)(16,23,25,22)(19,26,24,27);(4,5)(6,9)(7,8)(10,11)(12,16,17,25)(13,19,20,24)(14,22,15,23)(18,26,21,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55911);