Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1588 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)(5,7)(8,9)(10,11)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,26)(25,27);(4,5,6,7)(12,14)(13,17)(15,22)(16,23)(18,25)(19,21)(20,26)(24,27);(13,18)(16,24)(17,25)(19,20)(21,26)(23,27);(9,11)(14,22)(17,21)(23,27)(25,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25905);