Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#10980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C22×C3×#121 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5)(6,7)(12,13)(14,18,19,16)(15,17);(5,7)(8,9)(10,11)(12,14)(13,16)(15,19)(17,18);(9,11)(12,14)(13,16)(15,19)(17,18)156 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#10980, #72253393 |
| maximal subgroup of | C3×#449469, D3×#10980, #72253393 |
| maximal normal subgroup of | C3×#449469, D3×#10980, #72253393 |
| maximal quotient of | C3×#111516 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3968);