Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#15718 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1278 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)(8,9,12,11)(10,14,15,13)(16,17,20,19)(18,22,23,21)(25,26)(27,31);(10,15)(13,14)(17,19)(18,23)(24,25,28,26)(27,30,31,29);(8,10)(9,13)(11,14)(12,15)(16,18,20,23)(17,21,19,22)(24,26,28,25)(27,30,31,29);(4,5)(6,7)(8,11,12,9)(10,14,15,13)(16,19,20,17)(18,22,23,21)(24,27,28,31)(25,29,26,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8706);