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) = C24 |
| direct product | C3×#8398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
782 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)(4,5,9,10,8,6,7,11)(12,13,16,21,17,20,24,27)(14,19,22,26,15,18,23,25);(4,6,8,5)(7,11,9,10)(12,14,17,15)(13,18,20,19)(16,23,24,22)(21,26,27,25);(4,7,8,9)(5,10,6,11)(12,15,17,14)(13,19,20,18)(16,22,24,23)(21,25,27,26)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1386);