Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3514 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,12,13,19)(7,15,17,10)(9,16,18,14)(20,21,25,29)(22,30,33,34)(23,26,24,28)(27,32,35,31);(4,6)(5,9)(7,16)(8,13)(10,19)(11,18)(12,15)(14,17)(21,26)(23,24)(27,34)(28,29)(30,35)(31,32);(20,22)(21,27)(23,31)(24,32)(25,33)(26,34)(28,30)(29,35);(6,13)(9,18)(12,19)(14,16)(20,23,25,24)(21,26,29,28)(22,31,33,32)(27,34,35,30)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11583);