Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(12,13,18,22)(14,23,19,26)(15,24,27,25)(16,21,17,20);(6,10)(8,11)(12,14,18,19)(13,15,22,27)(16,24,17,25)(20,26,21,23);(4,5)(6,11)(7,9)(8,10)(12,15)(13,19)(14,22)(16,23)(17,26)(18,27)(20,24)(21,25);(12,16,18,17)(13,20,22,21)(14,24,19,25)(15,26,27,23)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12064);