Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#14908 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,7)(9,10)(12,13,16,15)(14,17,18,19)(20,21)(22,26)(23,27)(24,25);(8,9,11,10)(14,18)(17,19)(21,25)(22,27);(8,10,11,9)(12,14)(13,17)(15,19)(16,18)(20,22,24,27)(21,26,25,23);(4,5)(6,7)(12,15,16,13)(14,19,18,17)(20,23,24,26)(21,27,25,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7896);