Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#577 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1328 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)(10,15)(11,23)(12,14)(13,18)(16,24)(17,27)(19,30)(20,35)(21,28)(22,31)(25,32)(26,33)(29,34)(36,38)(37,39);(9,14)(10,19)(12,27)(16,23)(17,18)(20,34)(21,22)(24,25)(28,30)(32,33)(35,38)(36,37);(5,7)(8,10,13,22)(9,15,18,31)(11,20,26,36)(12,21,27,19)(14,28,17,30)(16,29,32,39)(23,35,33,38)(24,34,25,37);(8,11)(9,16)(10,20)(12,25)(13,26)(14,23)(15,29)(17,33)(18,32)(19,34)(21,37)(22,36)(24,27)(28,35)(30,38)(31,39)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17446);