Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×C4×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ C2×C3×C4×C8 ◁ G | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,5,6,7);(8,9,12,16,13,10,17,15)(11,14,19,22,20,18,23,21);(8,10)(9,13)(11,18)(12,15)(14,20)(16,17)(19,21)(22,23);(8,11,12,19,13,20,17,23)(9,14,16,22,10,18,15,21)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13497);