Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#7689 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8,9,10)(13,17)(14,15)(18,19)(20,21,24,26,25,22,23,27);(4,5,7,9)(6,10,11,8)(12,13,16,18)(14,17,15,19)(20,22,25,21)(23,26,24,27);(12,14,16,15)(13,17,18,19)(21,22)(26,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 677);