Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
526 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);(3,4,9,14)(5,15,18,13)(6,19)(7,23,16,12)(8,10,25,27)(11,28)(17,31)(20,33)(21,32)(22,26)(24,29)(30,34);(3,5,16,25,9,18,7,8)(4,10,23,13,14,27,12,15)(6,17,21,34,22,28,20,24)(11,26,30,32,31,19,29,33);(3,6)(4,11)(5,17)(7,20)(8,24)(9,22)(10,26)(12,29)(13,32)(14,31)(15,33)(16,21)(18,28)(19,27)(23,30)(25,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26624);