Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#15038 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,6)(9,12)(10,14)(16,17)(18,22)(19,20)(21,23)(24,25,27,29)(26,30,31,28);(4,5,7,6)(9,12)(13,15)(17,20)(21,22);(4,5,7,6)(8,9,11,12)(10,13,14,15)(16,17)(18,21)(19,20)(22,23)(24,26,27,31)(25,28,29,30);(4,6,7,5)(8,10)(9,13)(11,14)(12,15)(16,18,19,23)(17,21,20,22)(24,25)(26,28)(27,29)(30,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8026);