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×#25558 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1102 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)(16,17,22,26)(18,23,27,24)(19,28,29,31)(20,25,21,30);(8,10,11,15)(9,12,13,14)(16,18)(17,23)(19,20)(21,29)(22,27)(24,26)(25,28)(30,31);(4,5,6,7)(16,19,20,18,22,29,21,27)(17,24,30,31,26,23,25,28);(9,13)(12,14)(16,20,22,21)(17,25,26,30)(18,19,27,29)(23,28,24,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18546);