Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#33718 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,13,17)(10,18,20,32)(11,23,25,38)(12,26,24,16)(14,27,28,21)(15,30,31,19)(22,37,36,34)(29,35,33,39);(8,10,19,35,12,22,25,27,13,20,30,39,24,36,11,21)(9,14,23,34,16,29,31,32,17,28,38,37,26,33,15,18);(4,5,6,7);(10,20)(14,28)(18,32)(21,27)(22,36)(29,33)(34,37)(35,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26706);