Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#20722 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G | |
1866 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)(12,13)(14,18,19,16)(15,17);(5,8)(9,11);(12,14)(13,16)(15,19)(17,18);(4,6,7,10)(5,9,8,11)192 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#20722, #72271071, #72271213, … |
| maximal subgroup of | D3×#20722, #72271071, C3×#6454041, … |
| maximal normal subgroup of | D3×#20722, #72271071, C3×#6454041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13710);