Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#11682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#102 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C3×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,7,10,8,6,11,9)(12,13)(14,23,26,39)(15,19)(16,30)(17,22)(18,25,34,43)(20,28)(21,29)(24,37,27,40)(31,33,41,35)(32,36)(38,42);(4,6)(5,8)(7,9)(10,11)(12,14)(13,18)(15,27)(16,31)(17,26)(19,35)(20,37)(21,33)(22,34)(23,36)(24,29)(25,42)(28,43)(30,39)(32,41)(38,40);(5,6)(9,10)(12,15,17,29)(13,19,22,21)(14,24,26,27)(16,28,32,42)(18,33,34,35)(20,36,38,30)(23,37,39,40)(25,41,43,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4670);