Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#15332 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,7)(12,13,15,17)(14,18,19,16)(20,21)(22,26)(23,25)(24,27);(9,11)(20,21,23,25)(22,26,27,24);(8,9)(10,11)(12,14,15,19)(13,16,17,18)(20,21)(22,24)(23,25)(26,27);(4,5)(6,7)(12,13)(14,16)(15,17)(18,19)(20,22,23,27)(21,24,25,26)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8320);