Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C16 · Schur multiplier (H2) = C4 |
| direct product | C3×#7053 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C42 |
| commutator | C4 | C3×C4×C16 |
| Frattini, cosocle | C82 | C22×C3 |
| socle, prime radical | C22×C3 | #322 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,6,10,7,12,14,18,8,11,13,19,15,9,16,17);(5,9,11,12)(6,13)(10,18,19,17)(14,16)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 41);