Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#20617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
3610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9)(10,18)(11,15)(12,24)(13,16)(14,20)(17,27)(19,35)(21,32)(22,33)(23,30)(25,29)(26,31)(28,36)(34,37)(38,39);(8,10)(9,14)(11,19)(12,25)(13,21)(15,28)(16,29)(17,32)(18,24)(20,27)(22,38)(23,37)(26,36)(30,39)(31,34)(33,35);(8,11,13,23)(9,15,16,30)(10,19,21,37)(12,26,27,22)(14,28,29,39)(17,33,24,31)(18,34,32,35)(20,38,25,36);(9,16)(14,29)(15,30)(17,24)(18,32)(28,39)(31,33)(34,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13605);