Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3054 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,22)(12,27)(13,16)(14,19)(15,26)(17,25)(20,38)(21,35)(23,33)(24,34)(28,31)(29,37)(30,39)(32,36);(8,10,13,21)(9,14,16,29)(11,23,24,39)(12,28,25,20)(15,31,32,38)(17,33,27,30)(18,34,35,22)(19,36,37,26);(5,7)(11,24)(14,29)(15,32)(19,37)(22,34)(23,39)(26,36)(30,33)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6305);