Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24 |
| direct product | C3×#7879 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C23×C3 | C2×C4⋊C4 |
| prime radical | C24×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5)(6,7)(9,13)(10,11)(14,15)(16,17,19,22,20,23,26,29)(18,21,25,28,24,27,30,31);(5,7)(8,9,11,15,12,14,10,13)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(9,14)(13,15)(18,24)(21,27)(25,30)(28,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 867);