Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#34974 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,11)(8,9)(12,13,16,17)(14,18,15,19)(20,21,23,24)(22,25,27,26);(5,9)(10,11)(13,17)(18,19)(20,21)(22,26)(23,24)(25,27);(4,6,8,7)(5,10,9,11)(13,17)(18,19)(21,24)(25,26);(4,7,8,6)(5,11,9,10)(12,14,16,15)(13,18,17,19)(20,22,23,27)(21,25,24,26);(12,15,16,14)(13,19,17,18)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27962);