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×#40613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1778 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)(12,13,16,18)(14,15,19,17)(20,21,23,25)(22,26,27,24);(4,5)(6,10)(7,8)(9,11)(12,14,16,19)(13,17,18,15);(5,8)(9,10)(13,18)(15,17)(20,22,23,27)(21,24,25,26);(4,5)(6,9)(7,8)(10,11)(12,15)(13,14)(16,17)(18,19);(4,6)(5,9)(7,11)(8,10)(12,14)(13,15)(16,19)(17,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33601);