Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#25468 |
| 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 ◁ C3×C4×C4⋊C4 ◁ G | |
708 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)(4,5,7,6)(8,9,12,15)(10,14,11,13)(16,17,19,21)(18,24,26,22)(20,23,27,29)(25,30,28,31);(4,6,7,5)(8,10,12,11)(9,13,15,14)(17,21)(18,25)(20,27)(22,30)(24,31)(26,28);(5,6)(16,18,19,26)(17,22,21,24)(20,28,27,25)(23,30,29,31);(8,11,12,10)(9,14,15,13)(17,21)(18,26)(23,29)(25,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18456);