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 |
| direct product | C3×#36184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(5,7)(8,9,12,14)(10,11,13,15)(17,20)(21,23);(8,10)(9,11)(12,13)(14,15)(16,17)(18,21)(19,20)(22,23);(4,5,6,7)(8,11)(9,13)(10,14)(12,15)(16,17)(18,21)(19,20)(22,23);(8,10)(9,11)(12,13)(14,15)(16,18,19,22)(17,21,20,23);(8,11)(9,13)(10,14)(12,15)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29172);