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×#18657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13)(6,14,17,18)(7,11,8,12)(10,16,19,15)(20,21)(22,30)(23,26)(24,28)(25,29)(27,31)(32,35)(33,34);(4,6,9,17)(5,10,13,19)(7,15,8,16)(11,18,12,14)(21,26)(23,24)(27,34)(28,29)(30,35)(31,32);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,18,19,14)(20,22,25,33)(21,27,29,35)(23,31,24,32)(26,34,28,30);(4,7,9,8)(5,12,13,11)(6,16,17,15)(10,18,19,14)(20,23,25,24)(21,28,29,26)(22,31,33,32)(27,30,35,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11645);