Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33630 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13,16,19)(14,24,26,41)(15,23,18,21)(17,30,29,38)(20,36,34,27)(22,28,35,32)(25,42,33,39)(31,40,43,37);(4,6,7,11)(5,8,9,10)(13,19)(20,34)(21,23)(22,35)(24,41)(30,38)(37,40)(39,42);(12,14)(13,20)(15,25)(16,26)(17,31)(18,33)(19,34)(21,37)(22,39)(23,40)(24,30)(27,32)(28,36)(29,43)(35,42)(38,41);(12,15,16,18)(13,21,19,23)(14,25,26,33)(17,28,29,32)(20,37,34,40)(22,38,35,30)(24,39,41,42)(27,31,36,43)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26618);