Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1136 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)(10,14)(11,13)(12,15)(16,17,21,25)(18,26,28,30)(19,24,20,23)(22,27,31,29);(8,10,11,15)(9,12,13,14)(18,27)(19,20)(22,30)(23,24)(26,31)(28,29);(4,5,6,7)(16,18,19,29,21,28,20,27)(17,22,23,26,25,31,24,30);(9,13)(12,14)(18,28)(22,31)(26,30)(27,29)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18543);