Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#8378 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C2×C3×#553 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,8,10,9,6,7,11)(12,13)(14,20,22,17)(15,23,24,18)(16,19)(21,25)(26,27);(4,6)(5,9)(7,11)(8,10)(12,14)(13,17)(15,21)(16,22)(18,25)(19,20)(23,26)(24,27);(4,7,9,8)(5,10,6,11)(12,15,16,24)(13,18,19,23)(14,21,22,27)(17,25,20,26)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1366);