Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#22126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,7,6)(9,12)(10,14)(16,17,20,24)(18,21,25,29)(19,22,27,23)(26,28,31,30);(4,6,7,5)(8,9)(10,13)(11,12)(14,15)(16,18)(17,21)(19,26)(20,25)(22,28)(23,30)(24,29)(27,31);(9,12)(13,15)(17,22)(19,27)(21,28)(23,24)(26,31)(29,30);(8,10,11,14)(9,13,12,15)(18,25)(21,29)(26,31)(28,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15114);