Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#15051 |
| 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 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,8)(9,11)(12,13,15,17)(14,16,18,19)(20,21)(22,26)(23,27)(24,25)(29,30)(31,35);(5,8)(9,11)(21,25)(22,27)(28,29,32,30)(31,33,35,34);(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,22,24,27)(21,26,25,23)(28,30,32,29)(31,34,35,33);(4,6,7,10)(5,9,8,11)(12,14,15,18)(13,16,17,19)(20,23,24,26)(21,27,25,22)(28,31)(29,33)(30,34)(32,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8039);