Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#24623 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
718 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)(6,11)(7,9)(8,10)(12,13,17,19,15,20,16,21)(14,22,23,26,24,18,25,27);(4,6,8,7)(5,9,10,11)(12,14,15,24)(13,18,20,22)(16,25,17,23)(19,27,21,26);(4,7,8,6)(5,9,10,11)(13,19)(14,23)(16,17)(18,22)(20,21)(24,25);(5,10)(9,11)(13,20)(18,22)(19,21)(26,27)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17611);