Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3519 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,15)(8,10)(9,14)(11,17)(13,19)(16,18)(20,21,25,28)(22,30,31,34)(23,29,24,27)(26,33,35,32);(4,6)(5,9)(7,16)(8,13)(10,18)(11,19)(12,15)(14,17)(20,22)(21,26)(23,33)(24,32)(25,31)(27,34)(28,35)(29,30);(21,27)(23,24)(26,34)(28,29)(30,35)(32,33);(5,10)(6,13)(11,15)(14,16)(21,28)(22,31)(27,29)(32,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11616);