Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#15173 |
| 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 | |
1530 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)(14,18)(17,19)(20,21,24,26)(22,23,27,25);(13,16)(17,19)(20,22,24,27)(21,25,26,23);(8,9,11,10)(12,13)(14,17)(15,16)(18,19)(20,22)(21,23)(24,27)(25,26);(4,5,6,7)(8,10,11,9)(12,14)(13,17)(15,18)(16,19)(20,23)(21,22)(24,25)(26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8161);