Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C2×C3×#3028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
4790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(35,36,37);(3,4,8,12)(5,9,15,22)(6,10,17,24)(7,18,19,11)(13,25,27,20)(14,28,29,21)(16,30,31,23)(26,32,34,33);(3,5)(4,9)(6,13)(7,14)(8,15)(10,20)(11,21)(12,22)(16,26)(17,27)(18,28)(19,29)(23,32)(24,25)(30,33)(31,34);(3,6)(4,10)(5,13)(7,16)(8,17)(9,20)(11,23)(12,24)(14,26)(15,27)(18,30)(19,31)(21,32)(22,25)(28,33)(29,34);(3,7,8,19)(4,11,12,18)(5,14,15,29)(6,16,17,31)(9,21,22,28)(10,23,24,30)(13,26,27,34)(20,32,25,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53385);