Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60621 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,22)(15,17)(16,20)(18,21)(19,23)(24,26)(25,27);(5,8)(9,11)(12,14,17,25,18,24,16,23)(13,19,20,26,21,27,15,22);(4,6,7,10)(5,9,8,11)(12,15,16,21,18,20,17,13)(14,22,23,27,24,26,25,19);(12,16,18,17)(13,20,21,15)(14,23,24,25)(19,26,27,22);(14,24)(19,27)(22,26)(23,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53609);