Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
732 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)(4,5,8,10)(6,9,7,11)(12,13,18,22)(14,24,20,26)(15,25,16,19)(17,23,27,21);(4,6,8,7)(5,9,10,11)(12,14)(13,16)(15,22)(17,19)(18,20)(21,26)(23,24)(25,27);(4,7,8,6)(5,9,10,11)(12,15,18,16)(13,19,22,25)(14,23,20,21)(17,26,27,24);(12,16,18,15)(13,20,22,14)(17,26,27,24)(19,21,25,23)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19393);