Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#29724 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3)(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,23)(20,25)(21,24)(26,35)(27,37)(28,36)(29,38)(30,40)(31,39)(32,42)(33,43)(34,41)(44,46)(45,47);(9,12)(13,15)(16,18,21,28)(17,22,24,36)(19,29,31,45)(20,32,34,27)(23,38,39,47)(25,42,41,37)(26,33,44,30)(35,43,46,40);(5,7)(8,10)(9,13)(11,14)(12,15)(16,19,21,31)(17,23,24,39)(18,26,28,44)(20,33,34,30)(22,35,36,46)(25,43,41,40)(27,45,32,29)(37,47,42,38);(10,14)(13,15)(17,24)(22,36)(23,39)(25,41)(35,46)(37,42)(38,47)(40,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22712);