Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1088 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)(8,9,11,12)(10,14,15,13)(16,17,19,22)(18,24,25,31)(20,28,27,23)(21,29,30,26);(4,5)(6,7)(9,12)(13,14)(16,17)(18,21)(19,22)(20,23)(24,26)(25,30)(27,28)(29,31);(5,7)(16,18,20,26,19,25,27,29)(17,21,23,24,22,30,28,31);(8,10,11,15)(9,13,12,14)(17,22)(21,30)(23,28)(24,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18670);