Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#15695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1538 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)(5,7)(8,9)(10,13)(11,12)(14,15)(16,17,19,21)(18,22,23,20)(25,28)(26,30);(9,11)(10,14)(18,23)(20,22)(24,25,27,28)(26,31,30,29);(8,10,12,14)(9,13,11,15)(16,18,19,23)(17,20,21,22)(24,25)(26,29)(27,28)(30,31);(4,5)(6,7)(8,11,12,9)(10,15,14,13)(16,17)(18,20)(19,21)(22,23)(24,26,27,30)(25,29,28,31)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8683);