Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#12592 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C22 | C2×C3×C4×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3)(8,9)(10,13)(11,12)(14,15)(16,17,19,23,20,24,29,35,21,25,30,36,31,37,41,45)(18,22,27,33,28,34,40,44,26,32,38,42,39,43,46,47);(4,5,6,7)(9,12)(13,15)(16,18)(17,22)(19,27)(20,28)(21,26)(23,33)(24,34)(25,32)(29,40)(30,38)(31,39)(35,44)(36,42)(37,43)(41,46)(45,47);(8,10,11,14)(9,13,12,15)(18,26)(22,32)(27,38)(28,39)(33,42)(34,43)(40,46)(44,47)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5580);