Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ G | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 67 points:
(1,2,3)(4,5,11,17)(6,18,23,34)(7,24,27,41)(8,14,28,38)(9,16,10,15)(12,22,35,21)(13,31,37,26)(19,44,47,32)(20,43,48,58)(25,52,53,61)(29,40,30,39)(33,50,57,49)(36,55,60,54)(42,51,59,46)(45,63,64,56)(62,66,67,65);(4,6,10,22,11,23,9,21)(5,12,16,34,17,35,15,18)(7,19,31,51,27,47,26,46)(8,20,30,50,28,48,29,49)(13,32,41,59,37,44,24,42)(14,33,40,58,38,57,39,43)(25,45,55,66,53,64,54,65)(36,56,61,67,60,63,52,62);(4,7)(5,13)(6,19)(8,25)(9,26)(10,31)(11,27)(12,32)(14,36)(15,24)(16,41)(17,37)(18,42)(20,45)(21,46)(22,51)(23,47)(28,53)(29,54)(30,55)(33,56)(34,59)(35,44)(38,60)(39,52)(40,61)(43,62)(48,64)(49,65)(50,66)(57,63)(58,67);(4,8)(5,14)(6,20)(7,25)(9,29)(10,30)(11,28)(12,33)(13,36)(15,39)(16,40)(17,38)(18,43)(19,45)(21,49)(22,50)(23,48)(24,52)(26,54)(27,53)(31,55)(32,56)(34,58)(35,57)(37,60)(41,61)(42,62)(44,63)(46,65)(47,64)(51,66)(59,67);(8,28)(14,38)(20,48)(25,53)(29,30)(33,57)(36,60)(39,40)(43,58)(45,64)(49,50)(52,61)(54,55)(56,63)(62,67)(65,66)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53460);