Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#13936 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,6,7)(9,12)(13,15)(16,17)(18,22)(19,20)(21,23)(25,28)(26,30);(9,12)(13,15)(17,20)(21,22)(24,25,27,28)(26,29,30,31);(8,9,11,12)(10,13,14,15)(16,17)(18,21)(19,20)(22,23)(24,25)(26,29)(27,28)(30,31);(8,10,11,14)(9,13,12,15)(16,18,19,23)(17,21,20,22)(24,26)(25,29)(27,30)(28,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6924);