Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C2×C4⋊C8 |
| prime radical | C24×C3 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C23×C3×C8 ◁ G | |
698 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);(3,4,7,12,8,11,21,26,9,13,22,27,6,19,20,32)(5,14,17,29,18,24,31,33,15,28,30,34,16,10,23,25);(3,5,6,16,9,15,8,18)(4,10,11,14,13,24,19,28)(7,17,20,23,22,30,21,31)(12,25,26,29,27,33,32,34);(5,15)(10,24)(14,28)(16,18)(17,30)(23,31)(25,33)(29,34)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26388);