Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#14184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,6)(12,13)(14,16)(15,18)(17,19)(20,21,23,25)(22,24,26,27);(9,11)(13,14)(17,18)(20,21,23,25)(22,24,26,27);(8,9)(10,11)(12,14,16,13)(15,18,19,17)(20,21)(22,24)(23,25)(26,27);(4,6,7,5)(12,15,16,19)(13,17,14,18)(20,22,23,26)(21,24,25,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7172);