Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#24667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
912 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)(10,14)(11,12)(13,15)(16,17,21,23,19,24,20,25)(18,26,29,31,27,22,28,30);(4,5)(6,7)(9,12)(13,14)(16,18)(17,22)(19,27)(20,28)(21,29)(23,30)(24,26)(25,31);(5,7)(17,23)(20,21)(22,30)(24,25)(26,31)(28,29);(8,10,11,15)(9,13,12,14)(17,24)(18,27)(23,25)(28,29)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17655);