Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C24×C3 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
552 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);(3,4,8,12)(5,13,14,31)(6,17,18,34)(7,22,20,11)(9,24,25,15)(10,27,28,21)(16,29,23,32)(19,26,33,30);(6,18)(10,28)(13,31)(15,24)(17,34)(19,33)(21,27)(29,32);(3,5,7,23,8,14,20,16)(4,9,11,30,12,25,22,26)(6,19,21,24,18,33,27,15)(10,29,17,31,28,32,34,13)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26643);