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×#2212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
706 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)(5,13)(6,18)(7,22)(8,24)(9,14)(10,16)(11,15)(12,23)(17,31)(19,30)(20,34)(21,26)(25,27)(28,32)(29,33);(3,5,7,16,9,17,23,8)(4,10,12,13,14,24,22,31)(6,15,19,32,21,33,20,25)(11,26,27,30,29,18,28,34);(3,6)(4,11)(5,15)(7,19)(8,25)(9,21)(10,26)(12,27)(13,30)(14,29)(16,32)(17,33)(18,24)(20,23)(22,28)(31,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26579);