Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 67 points:
(1,2,3)(4,5,12,17)(6,18,22,39)(7,8,25,13)(9,14,31,43)(10,32,35,48)(11,16,36,19)(15,44,45,34)(20,40,50,62)(21,27,33,38)(23,26,28,37)(24,46,53,59)(29,56,58,67)(30,42,51,47)(41,64,66,57)(49,54,60,61)(52,63,55,65);(4,6,11,7)(5,8,16,18)(9,23,30,20)(10,33,34,53)(12,22,36,25)(13,19,39,17)(14,40,42,26)(15,38,32,59)(21,44,24,35)(27,48,46,45)(28,51,50,31)(29,55,57,60)(37,43,62,47)(41,65,56,61)(49,58,52,64)(54,66,63,67);(4,7,11,6)(5,13,16,39)(8,19,18,17)(9,28,30,50)(10,24,34,21)(12,25,36,22)(14,26,42,40)(15,38,32,59)(20,31,23,51)(27,48,46,45)(29,55,57,60)(33,35,53,44)(37,47,62,43)(41,54,56,63)(49,58,52,64)(61,67,65,66);(4,8,12,13)(5,7,17,25)(6,19,22,16)(9,26,31,37)(10,27,35,38)(11,18,36,39)(14,23,43,28)(15,24,45,53)(20,47,50,42)(21,48,33,32)(29,54,58,61)(30,40,51,62)(34,46,44,59)(41,52,66,55)(49,67,60,56)(57,63,64,65);(4,9,11,30)(5,14,16,42)(6,20,7,23)(8,26,18,40)(10,29,34,57)(12,31,36,51)(13,37,39,62)(15,41,32,56)(17,43,19,47)(21,49,24,52)(22,50,25,28)(27,54,46,63)(33,60,53,55)(35,58,44,64)(38,61,59,65)(45,66,48,67);(4,10)(5,15)(6,21)(7,24)(8,27)(9,29)(11,34)(12,35)(13,38)(14,41)(16,32)(17,45)(18,46)(19,48)(20,49)(22,33)(23,52)(25,53)(26,54)(28,55)(30,57)(31,58)(36,44)(37,61)(39,59)(40,63)(42,56)(43,66)(47,67)(50,60)(51,64)(62,65)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55967);