Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#10985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9)(10,18)(11,22,24,15)(12,25)(13,17)(14,20)(16,27)(19,33,35,32)(21,31)(23,30,39,37)(26,29)(28,34,36,38);(4,5)(6,7)(8,10,13,21)(9,14,17,29)(11,19,24,35)(12,26,27,20)(15,28,22,36)(16,31,25,18)(23,38,39,34)(30,33,37,32);(5,7)(8,11)(9,15)(10,19)(12,23)(13,24)(14,28)(16,30)(17,22)(18,32)(20,34)(21,35)(25,37)(26,38)(27,39)(29,36)(31,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3973);