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 | C22×C3×#359 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C23×C3 | #102 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(37,38,39);(3,4);(5,6,9,14,11,15,10,25)(7,16,18,32,19,33,26,30)(8,20,22,31,24,35,23,13)(12,21,27,34,29,36,28,17);(5,7)(6,12)(8,21)(9,18)(10,26)(11,19)(13,30)(14,27)(15,29)(16,20)(17,23)(22,34)(24,36)(25,28)(31,32)(33,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26371);