Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G | |
814 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)(10,15)(11,13)(12,14)(16,17)(18,25)(19,24)(20,30)(21,27)(22,26)(23,28)(29,31);(4,5)(6,7)(8,10,12,11)(9,13,14,15)(16,18,21,19,23,29,22,20)(17,24,26,25,28,30,27,31);(5,7)(8,11,12,10)(9,13,14,15)(16,19,21,29,23,20,22,18)(17,25,26,30,28,31,27,24);(9,14)(13,15)(16,20,22,29,23,19,21,18)(17,25,27,24,28,31,26,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19812);