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×#15057 |
| 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)(6,10)(12,13,16,14)(15,18,19,17)(21,22)(23,27)(30,34)(33,35);(13,14)(17,18)(20,21,24,22)(23,25,27,26)(28,29,32,31)(30,33,34,35);(4,5)(6,9)(7,8)(10,11)(12,14,16,13)(15,18,19,17)(20,22,24,21)(23,26,27,25)(28,30)(29,33)(31,35)(32,34);(4,6,7,10)(5,9,8,11)(12,15,16,19)(13,17,14,18)(20,23)(21,25)(22,26)(24,27)(28,31,32,29)(30,35,34,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8045);