Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#16249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
4102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,7)(9,11)(13,15);(8,9)(10,11)(17,19);(9,11)(12,13)(14,15)(16,17)(18,19);(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)156 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#16249, #72248160, #72253747, … |
| maximal subgroup of | D3×#16249, #72248160, C3×#846041, … |
| maximal normal subgroup of | D3×#16249, #72248160, C3×#846041, … |
| maximal quotient of | C3×#454002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9237);