Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#13270 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #124 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1985 |
| prime radical | C23×C3 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,9,15,10,14,26,29,11,16,27,33,8,22,18,34)(6,17,21,25,20,30,35,23,19,12,28,24,13,31,32,7);(4,6,18,32,8,13,27,28,11,19,26,35,10,20,9,21)(5,12,29,7,22,30,15,24,16,17,34,23,14,31,33,25);(4,7,8,24,11,23,10,25)(5,13,14,6,16,20,22,19)(9,17,18,31,27,12,26,30)(15,32,29,21,33,35,34,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6258);