Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18125 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
928 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)(4,5,8,11)(6,12,14,19)(7,15,17,10)(9,16,18,13)(25,31)(32,35);(4,6,8,14)(5,9,11,18)(7,16,17,13)(10,19,15,12)(20,21)(22,25)(23,26)(24,27)(28,32)(29,31)(30,33)(34,35);(20,22)(21,25)(23,28)(24,29)(26,32)(27,31)(30,34)(33,35);(20,23,24,30)(21,26,27,33)(22,28,29,34)(25,32,31,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11113);