Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#21184 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#389 |
| prime radical | C24×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13,10,14,8,24)(6,15,17,11,19,26,18,27)(7,20,23,28,21,34,22,12)(16,25,31,33,30,35,32,29);(4,6,9,17,10,19,8,18)(5,11,13,26,14,27,24,15)(7,16,23,31,21,30,22,32)(12,25,20,33,28,35,34,29);(4,7,8,22,10,21,9,23)(5,12,13,20,14,28,24,34)(6,16,18,32,19,30,17,31)(11,25,26,33,27,35,15,29);(7,21)(12,28)(16,30)(20,34)(22,23)(25,35)(29,33)(31,32)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14172);