Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(3,4,10,16)(5,14,19,25)(6,20,22,33)(7,15,9,13)(8,11,18,26)(12,27,29,17)(21,34,24,30)(23,31,32,28);(3,5,9,18,10,19,7,8)(4,11,15,25,16,26,13,14)(6,17,24,32,22,27,21,23)(12,20,31,34,29,33,28,30);(3,6)(4,12)(5,17)(7,21)(8,23)(9,24)(10,22)(11,20)(13,28)(14,30)(15,31)(16,29)(18,32)(19,27)(25,34)(26,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26581);