Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5,11,17)(6,18,23,37)(7,24,29,41)(8,30,33,14)(9,16,10,15)(12,22,38,21)(13,27,42,28)(19,35,49,52)(20,43,50,46)(25,54,48,39)(26,53,58,64)(31,36,51,62)(32,44,34,45)(40,57,59,56)(47,61,66,67)(55,65,60,63);(4,6,11,23)(5,12,17,38)(7,25,29,48)(8,31,33,51)(9,22,10,21)(13,39,42,54)(14,43,30,46)(15,37,16,18)(19,28,49,27)(20,34,50,32)(24,52,41,35)(26,47,58,66)(36,45,62,44)(40,61,59,67)(53,63,64,65)(55,57,60,56);(4,7)(5,13)(6,19)(8,26)(9,27)(10,28)(11,29)(12,35)(14,40)(15,41)(16,24)(17,42)(18,39)(20,47)(21,25)(22,48)(23,49)(30,59)(31,60)(32,56)(33,58)(34,57)(36,61)(37,54)(38,52)(43,65)(44,53)(45,64)(46,63)(50,66)(51,55)(62,67);(4,8)(5,14)(6,20)(7,26)(9,34)(10,32)(11,33)(12,36)(13,40)(15,45)(16,44)(17,30)(18,46)(19,47)(21,51)(22,31)(23,50)(24,53)(25,55)(27,57)(28,56)(29,58)(35,61)(37,43)(38,62)(39,63)(41,64)(42,59)(48,60)(49,66)(52,67)(54,65)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23431);