Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#13262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #459 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C23×C3 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
462 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)(5,7)(8,9,12,17,13,28,26,35,15,19,25,34,14,18,11,24)(10,20,21,31,29,36,39,33,23,37,38,32,22,30,27,16);(4,5,6,7)(8,10,15,23)(9,16,19,33)(11,21,26,38)(12,27,25,39)(13,22,14,29)(17,30,34,36)(18,31,28,32)(20,35,37,24);(8,11,14,25,15,26,13,12)(9,17,28,35,19,34,18,24)(10,21,29,39,23,38,22,27)(16,30,32,37,33,36,31,20)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6250);