Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9)(10,18)(11,22)(12,27)(13,15)(14,19)(16,25)(17,24)(20,38)(21,35)(23,33)(26,34)(28,32)(29,37)(30,39)(31,36);(4,5,6,7);(8,10,13,21)(9,14,15,29)(11,23,26,39)(12,28,24,20)(16,32,31,38)(17,33,27,30)(18,34,35,22)(19,36,37,25);(9,15)(14,29)(16,31)(17,27)(18,35)(22,34)(30,33)(32,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26569);