Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#17574 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
1338 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)(6,8)(7,10)(9,11)(12,13)(14,20,22,17)(15,19)(16,18)(21,26,27,24)(23,25);(5,8)(6,11)(9,10)(12,14)(13,17)(15,21)(16,22)(18,20)(19,24)(23,27)(25,26);(13,18)(17,20)(19,25)(24,26);(12,15,16,23)(13,19,18,25)(14,21,22,27)(17,24,20,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10562);