Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41449 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1898 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)(4,5)(6,10)(7,9)(8,11)(12,13,17,15)(14,18,19,16)(21,24)(25,27);(4,5,7,9)(6,8,11,10)(12,14)(13,18)(15,16)(17,19);(6,11)(8,10)(13,15)(14,19)(20,21)(22,25)(23,24)(26,27);(4,6,7,11)(5,8,9,10)(12,15,17,13)(14,16,19,18)(20,22,23,26)(21,25,24,27);(4,5)(6,8)(7,9)(10,11)(12,16)(13,14)(15,19)(17,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34437);