Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C26×C4 |
| direct product | C3×#37150 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1880 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)(6,11)(7,9)(8,10)(13,17)(15,18)(21,24)(25,27);(5,7)(10,11)(13,17)(14,19);(4,6,9,8)(5,10,7,11)(12,13)(14,18)(15,19)(16,17)(20,21)(22,25)(23,24)(26,27);(4,7,9,5)(6,11,8,10)(12,14,16,19)(13,18,17,15);(4,8,9,6)(5,11,7,10)(12,15,16,18)(13,19,17,14)(20,22,23,26)(21,25,24,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30138);