Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10760 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #117 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #380 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×M16 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,12,19,14,18,13,17)(10,20,22,15,24,31,23,30)(11,25,27,33,28,16,26,32)(21,35,37,29,38,34,36,39);(5,7)(8,10,13,23,14,24,12,22)(9,15,19,31,18,30,17,20)(11,21,27,37,28,38,26,36)(16,29,33,35,25,39,32,34);(8,11)(9,16)(10,21)(12,26)(13,27)(14,28)(15,29)(17,32)(18,25)(19,33)(20,34)(22,36)(23,37)(24,38)(30,39)(31,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3748);