Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9664 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9,13,14,12,10,11,15)(16,17)(18,26)(19,22)(20,29)(21,25)(23,27)(24,30)(28,31);(4,5,6,7)(8,10,13,15,12,9,11,14)(16,18,20,19,21,27,30,28)(17,22,24,26,25,31,29,23);(8,11,12,13)(9,14,10,15)(16,19,20,27,21,28,30,18)(17,23,24,22,25,26,29,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2652);