Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3)(4,5)(6,20)(7,27)(8,18)(9,34)(10,16)(11,15)(12,19)(13,22)(14,30)(17,36)(21,53)(23,40)(24,41)(25,43)(26,42)(28,39)(29,46)(31,51)(32,64)(33,48)(35,45)(37,47)(38,63)(44,56)(49,57)(50,59)(52,55)(54,65)(58,66)(60,62)(61,67);(4,6,12,26)(5,13,19,43)(7,28,32,58)(8,25,11,22)(9,24,36,57)(10,23,33,59)(14,44,47,53)(15,42,18,20)(16,49,48,41)(17,50,34,40)(21,35,56,31)(27,62,64,52)(29,55,38,60)(30,54,37,61)(39,63,66,46)(45,67,51,65);(4,7,8,29,12,32,11,38)(5,14,15,45,19,47,18,51)(6,21,22,54,26,56,25,61)(9,35,33,30,36,31,10,37)(13,39,20,52,43,66,42,62)(16,46,17,27,48,63,34,64)(23,58,57,55,59,28,24,60)(40,44,41,65,50,53,49,67)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6406);