Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#9919 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,21)(19,22)(20,23)(24,27)(25,28)(26,30)(29,31);(9,13)(10,11)(14,15)(16,18,20,25)(17,21,23,28)(19,24,26,31)(22,29,30,27);(5,7)(8,10,12,11)(9,14,15,13)(16,19)(17,22)(18,24)(20,26)(21,27)(23,30)(25,31)(28,29)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2907);