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×#62980 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6114 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)(6,7,21,24)(8,25,20,13)(9,14,34,44)(10,35,38,54)(11,16,27,26)(15,45,47,37)(18,22,41,31)(19,40,36,46)(23,29,51,49)(28,52,30,39)(32,59,61,67)(33,43,55,53)(42,64,65,60)(48,63,62,66)(50,56,58,57);(4,6,11,20)(5,13,16,7)(8,12,21,27)(9,30,33,18)(10,36,37,29)(14,22,43,39)(15,46,35,51)(17,25,26,24)(19,45,49,38)(23,47,40,54)(28,55,41,34)(31,53,52,44)(32,56,60,62)(42,58,59,66)(48,61,57,64)(50,67,63,65);(4,7)(5,6)(8,26)(9,31)(10,23)(11,13)(12,24)(14,41)(15,19)(16,20)(17,21)(18,44)(22,34)(25,27)(28,43)(29,54)(30,53)(32,58)(33,52)(35,49)(36,47)(37,40)(38,51)(39,55)(42,62)(45,46)(48,65)(50,61)(56,59)(57,67)(60,66)(63,64);(4,8,27,6)(5,13,26,24)(7,17,25,16)(9,28,55,18)(10,29,45,19)(11,21,12,20)(14,39,53,31)(15,40,54,51)(22,44,52,43)(23,47,46,35)(30,33,41,34)(32,56,64,48)(36,38,49,37)(42,63,67,58)(50,65,66,59)(57,60,62,61);(4,9,11,33)(5,14,16,43)(6,18,20,30)(7,22,13,39)(8,28,21,41)(10,32,37,60)(12,34,27,55)(15,42,35,59)(17,44,26,53)(19,48,49,57)(23,50,40,63)(24,31,25,52)(29,56,36,62)(38,61,45,64)(46,66,51,58)(47,65,54,67);(4,10)(5,15)(6,19)(7,23)(8,29)(9,32)(11,37)(12,38)(13,40)(14,42)(16,35)(17,47)(18,48)(20,49)(21,36)(22,50)(24,51)(25,46)(26,54)(27,45)(28,56)(30,57)(31,58)(33,60)(34,61)(39,63)(41,62)(43,59)(44,65)(52,66)(53,67)(55,64)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55968);