Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#24149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13)(14,19)(15,17)(16,20)(18,25)(21,26)(22,27)(23,24);(5,8)(9,10)(12,14)(13,19)(15,24)(16,22)(17,23)(18,26)(20,27)(21,25);(12,15,17,13)(14,21,23,27)(16,25,18,20)(19,26,24,22);(4,6,7,11)(5,9,8,10)(12,16,17,18)(13,20,15,25)(14,22,23,26)(19,27,24,21)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17137);