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×#42976 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2058 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)(12,13,17,15)(14,19,16,18)(20,21,24,25)(22,27,26,23);(4,5,8,6)(7,10,11,9)(14,16)(18,19)(21,25)(22,26);(5,6)(9,10)(12,14,17,16)(13,18,15,19)(20,22,24,26)(21,23,25,27);(4,6,8,5)(7,10,11,9)(12,15,17,13)(14,18,16,19)(20,23)(21,26)(22,25)(24,27);(4,7,8,11)(5,9,6,10)(12,16,17,14)(13,18,15,19)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35964);