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×#13339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G | |
534 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)(10,22,27,57)(11,28,34,47)(12,35,40,53)(13,20,14,19)(16,41,45,24)(17,30,50,31)(18,38,54,39)(23,42,61,70)(25,51,26,56)(29,66,60,46)(32,52,33,65)(36,43,63,44)(37,48,68,49)(55,58,71,59)(62,64,67,69);(4,5)(6,7)(8,10,15,27)(9,16,21,45)(11,29,34,60)(12,36,40,63)(13,26,14,25)(17,46,50,66)(18,51,54,56)(19,44,20,43)(22,53,57,35)(23,33,61,32)(24,38,41,39)(28,64,47,69)(30,58,31,59)(37,67,68,62)(42,49,70,48)(52,71,65,55);(5,7)(8,11)(9,17)(10,23)(12,37)(13,33)(14,32)(15,34)(16,42)(18,52)(19,49)(20,48)(21,50)(22,55)(24,62)(25,60)(26,29)(27,61)(28,35)(30,38)(31,39)(36,59)(40,68)(41,67)(43,66)(44,46)(45,70)(47,53)(51,64)(54,65)(56,69)(57,71)(58,63)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6327);