Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#12993 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C24×C3 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,7)(8,9)(10,11)(12,13,15,20,24,27,23,18,17,22,14,19,16,21,25,26);(4,5,6,7)(13,18)(14,23)(15,25)(16,24)(19,21)(20,27)(22,26);(9,11)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5981);