Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#34366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,6,7)(9,12)(10,14)(16,17,20,21)(18,24,25,22)(19,27,28,23)(26,29,31,30);(9,12)(13,15)(17,21)(18,25)(23,27)(26,31);(8,9,11,12)(10,13,14,15)(16,17)(18,22)(19,23)(20,21)(24,25)(26,29)(27,28)(30,31);(8,10)(9,13)(11,14)(12,15)(16,18,20,25)(17,22,21,24)(19,26,28,31)(23,29,27,30);(16,19)(17,23)(18,26)(20,28)(21,27)(22,29)(24,30)(25,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27354);