Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C2×C3×#3022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
4742 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);(3,4,9,14)(5,15,18,27)(6,11,20,26)(7,13,8,12)(10,21,24,17)(16,30,33,29)(19,28,22,25)(23,31,34,32);(3,5)(4,10)(6,16)(7,17)(8,21)(9,18)(11,23)(12,15)(13,27)(14,24)(19,31)(20,33)(22,32)(25,29)(26,34)(28,30);(3,6)(4,11)(5,16)(7,19)(8,22)(9,20)(10,23)(12,25)(13,28)(14,26)(15,29)(17,31)(18,33)(21,32)(24,34)(27,30);(3,7,9,8)(4,12,14,13)(5,17,18,21)(6,19,20,22)(10,15,24,27)(11,25,26,28)(16,31,33,32)(23,29,34,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53374);