Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
366 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)(4,5,6,7)(8,9)(10,14)(11,12)(13,15)(16,17)(18,28)(19,25)(20,35)(21,37)(22,27)(23,30)(24,33)(26,36)(29,42)(31,38)(32,44)(34,41)(39,46)(40,45)(43,47);(5,7)(9,12)(13,14)(16,18,19,29,21,31,33,45,22,32,34,46,36,47,20,30)(17,23,35,43,26,39,41,44,27,40,24,38,37,42,25,28);(8,10,11,15)(9,13,12,14)(16,19,21,33,22,34,36,20)(17,24,26,25,27,35,37,41)(18,29,31,45,32,46,47,30)(23,38,39,28,40,43,42,44)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6269);