Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23391 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
2514 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)(4,5)(6,7)(12,13)(14,16)(15,17)(18,19);(8,9)(10,11)(13,16)(14,19)(17,18);(5,7)(9,11);(13,17)(16,18)156 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#23391, #72248220 |
| maximal subgroup of | D3×#23391, C3×#394080, #72248220 |
| maximal normal subgroup of | D3×#23391, C3×#394080, #72248220 |
| maximal quotient of | C3×#234591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16379);