Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3)(8,9,15,21)(10,22,27,53)(11,28,31,50)(12,32,37,48)(13,20,14,19)(16,39,43,24)(17,38,45,30)(18,35,49,36)(23,54,57,71)(25,46,26,52)(29,47,34,44)(33,41,59,42)(40,63,70,66)(51,60,68,56)(55,61,58,69)(62,67,65,64);(4,5)(6,7)(8,10,15,27)(9,16,21,43)(11,23,31,57)(12,33,37,59)(13,26,14,25)(17,40,45,70)(18,46,49,52)(19,42,20,41)(22,48,53,32)(24,35,39,36)(28,61,50,69)(29,63,34,66)(30,60,38,56)(44,71,47,54)(51,62,68,65)(55,67,58,64);(5,7)(8,11)(9,17)(10,23)(12,34)(13,38)(14,30)(15,31)(16,40)(18,47)(19,50)(20,28)(21,45)(22,51)(24,58)(25,60)(26,56)(27,57)(29,37)(32,65)(33,66)(35,64)(36,67)(39,55)(41,61)(42,69)(43,70)(44,49)(46,54)(48,62)(52,71)(53,68)(59,63)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6310);