Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1742 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C2×C22⋊C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(23,24,25);(7,8,11,16,12,15,10,14)(9,17,18,22,19,13,20,21);(3,4,5,6)(7,9)(8,13)(10,18)(11,20)(12,19)(14,21)(15,17)(16,22);(7,10,12,11)(8,14,15,16)(9,18,19,20)(13,21,17,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13345);