Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25145 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,27,13,18,12,17)(10,15,28,39,21,31,20,30)(11,22,24,16,26,33,25,34)(19,35,36,29,23,38,37,32);(5,7)(8,10,12,20,13,21,14,28)(9,15,17,30,18,31,27,39)(11,23,25,36,26,19,24,37)(16,32,22,38,34,29,33,35);(8,11)(9,16)(10,19)(12,24)(13,26)(14,25)(15,29)(17,33)(18,34)(20,36)(21,23)(22,27)(28,37)(30,38)(31,32)(35,39);(5,7)(8,12,13,14)(9,17,18,27)(10,20,21,28)(11,24,26,25)(15,30,31,39)(16,33,34,22)(19,36,23,37)(29,38,32,35)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18133);