Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
1128 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)(5,13,15,31)(6,17,19,33)(7,22,20,11)(9,24,25,14)(10,27,29,21)(16,32,23,28)(18,30,34,26);(3,5)(4,9)(6,18)(7,23)(8,15)(10,28)(11,30)(12,25)(13,17)(14,21)(16,20)(19,34)(22,26)(24,27)(29,32)(31,33);(6,19)(10,29)(13,31)(14,24)(17,33)(18,34)(21,27)(28,32)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26517);