Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×#63093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#585 ◁ G | |
10698 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)(4,5,10,15)(6,16,20,37)(7,12,22,33)(8,13,24,35)(9,26,28,14)(11,27,31,19)(17,38,41,58)(18,39,43,59)(21,32,45,56)(23,48,47,34)(25,51,50,36)(29,49,54,42)(30,52,55,44)(40,60,61,67)(46,64,63,57)(53,65,66,62);(4,6,10,20)(5,11,15,31)(7,17,22,41)(8,18,24,43)(9,27,28,19)(12,29,33,54)(13,30,35,55)(14,37,26,16)(21,40,45,61)(23,49,47,42)(25,52,50,44)(32,53,56,66)(34,58,48,38)(36,59,51,39)(46,65,63,62)(57,67,64,60);(4,7)(5,12)(6,17)(8,21)(9,23)(10,22)(11,29)(13,32)(14,34)(15,33)(16,38)(18,40)(19,42)(20,41)(24,45)(25,46)(26,48)(27,49)(28,47)(30,53)(31,54)(35,56)(36,57)(37,58)(39,60)(43,61)(44,62)(50,63)(51,64)(52,65)(55,66)(59,67);(4,8)(5,13)(6,18)(7,21)(9,25)(10,24)(11,30)(12,32)(14,36)(15,35)(16,39)(17,40)(19,44)(20,43)(22,45)(23,46)(26,51)(27,52)(28,50)(29,53)(31,55)(33,56)(34,57)(37,59)(38,60)(41,61)(42,62)(47,63)(48,64)(49,65)(54,66)(58,67);(8,24)(13,35)(18,43)(21,45)(25,50)(30,55)(32,56)(36,51)(39,59)(40,61)(44,52)(46,63)(53,66)(57,64)(60,67)(62,65);(7,22)(12,33)(17,41)(21,45)(23,47)(29,54)(32,56)(34,48)(38,58)(40,61)(42,49)(46,63)(53,66)(57,64)(60,67)(62,65)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56081);