Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#25474 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,9,12,15)(10,14,11,13)(16,17,21,30,19,24,20,25)(18,26,29,23,27,22,28,31);(4,5)(6,7)(8,10,12,11)(9,13,15,14)(16,18,21,29,19,27,20,28)(17,22,30,31,24,26,25,23);(5,7)(16,18,19,27)(17,23,24,31)(20,29,21,28)(22,30,26,25);(8,11,12,10)(9,14,15,13)(17,24)(22,26)(23,31)(25,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18462);