Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#35306 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(5,7)(9,11)(12,13,17,19)(14,20,21,15)(16,24,25,18)(22,23,27,26);(8,9,10,11)(14,21)(15,20)(22,27)(23,26);(4,5)(6,7)(12,14)(13,15)(16,22)(17,21)(18,23)(19,20)(24,26)(25,27);(12,15)(13,14)(16,23)(17,20)(18,22)(19,21)(24,27)(25,26);(12,16,17,25)(13,18,19,24)(14,22,21,27)(15,23,20,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28294);