Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#1373 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | D8 |
| commutator | C8 | C22×C3×C8 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C22×C3 | #335 |
| prime radical | C23×C3 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(35,36,37);(3,4,7,12,8,14,6,22,9,15,17,11,23,13,24,29)(5,16,19,26,20,31,18,10,21,25,32,27,33,30,34,28);(3,5,17,32,8,20,24,34,9,21,7,19,23,33,6,18)(4,10,22,30,13,26,12,25,15,28,29,31,14,27,11,16)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4998);