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) = C26×C4 |
| direct product | C3×#36176 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,8,7)(6,9,11,10)(12,13,16,18)(14,15,17,19)(21,22)(26,27);(5,7)(9,10)(12,14)(13,15)(16,17)(18,19)(20,21,25,22)(23,26,24,27);(4,5,8,7)(6,10,11,9)(12,15)(13,17)(14,18)(16,19)(20,22,25,21)(23,27,24,26);(4,6,8,11)(5,9,7,10)(12,14)(13,15)(16,17)(18,19)(20,23,25,24)(21,26,22,27);(4,7,8,5)(6,9,11,10)(12,15)(13,17)(14,18)(16,19)(20,24,25,23)(21,27,22,26)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29164);