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×#15064 |
| 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 | |
1066 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)(14,17)(15,16)(18,19)(20,21,24,23)(22,26,27,25)(29,30)(31,35);(5,8)(9,11)(13,16)(17,19)(28,29,32,30)(31,33,35,34);(4,5,7,8)(6,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,22,24,27)(21,25,23,26)(28,30,32,29)(31,34,35,33);(4,6,7,10)(5,9,8,11)(12,14,15,18)(13,17,16,19)(20,23,24,21)(22,26,27,25)(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, 8052);