Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#2403 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C25×C3 ◁ G | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(16,19);(5,8)(9,11);(4,6,7,10)(5,9,8,11)(12,13)(14,16)(15,17)(18,19);(12,14)(13,16)(15,18)(17,19)147 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#33020, #72248899, #72270502, … |
| maximal subgroup of | C3×#6470225, D3×#33020, #72248899, … |
| maximal normal subgroup of | C3×#6470225, D3×#33020, #72248899, … |
| maximal quotient of | C3×#512119 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26008);