Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1186 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)(10,18)(11,15)(12,24)(13,17)(14,20)(16,26)(19,34)(21,33)(22,31)(23,30)(25,29)(27,35)(28,38)(32,36)(37,39);(8,10)(9,14,17,28)(11,19)(12,20)(13,21)(15,27,30,39)(16,18,24,33)(22,35)(23,36)(25,37)(26,38)(29,34,31,32);(8,11,13,23)(9,15,17,30)(10,19,21,36)(12,25,26,22)(14,27,28,39)(16,31,24,29)(18,32,33,34)(20,37,38,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3729);