Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C3×#36179 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(5,7)(9,10)(12,13,16,18)(14,15,17,19);(8,9,11,10)(12,14)(13,15)(16,17)(18,19);(4,5,6,7)(8,10,11,9)(12,15)(13,17)(14,18)(16,19);(12,14)(13,15)(16,17)(18,19);(12,15)(13,17)(14,18)(16,19)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29167);