Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2138 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C23×C3 | C2×D8 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(39,40,41);(3,4,5,6)(7,8)(9,18)(10,15)(11,23)(12,27)(13,17)(14,20)(16,25)(19,34)(21,35)(22,33)(24,29)(26,37)(28,32)(30,38)(31,36);(7,9)(8,14)(10,19)(11,24)(12,28)(13,21)(15,29)(16,32)(17,31)(18,23)(20,25)(22,36)(26,38)(27,30)(33,34)(35,37);(7,10,13,12)(8,15,17,27)(9,19,21,28)(11,22,26,25)(14,29,31,30)(16,23,33,37)(18,34,35,32)(20,24,36,38)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26364);