Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1090 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)(4,5,10,15)(6,16,19,45)(7,12,21,36)(8,24,29,59)(9,30,27,14)(11,32,34,18)(13,39,42,28)(17,46,47,44)(20,51,31,40)(22,53,26,58)(23,56,52,38)(25,43,61,35)(33,50,63,48)(37,62,41,55)(49,67,57,65)(54,66,60,64);(4,6)(5,11)(7,17)(8,25)(9,31)(10,19)(12,33)(13,40)(14,43)(15,34)(16,24)(18,28)(20,27)(21,47)(22,54)(23,57)(26,60)(29,61)(30,35)(32,39)(36,63)(37,65)(38,66)(41,67)(42,51)(44,58)(45,59)(46,53)(48,55)(49,52)(50,62)(56,64);(4,7)(5,12)(6,17)(8,26)(9,23)(10,21)(11,33)(13,41)(14,38)(15,36)(16,44)(18,50)(19,47)(20,49)(22,29)(24,58)(25,60)(27,52)(28,62)(30,56)(31,57)(32,48)(34,63)(35,64)(37,42)(39,55)(40,67)(43,66)(45,46)(51,65)(53,59)(54,61);(7,21)(12,36)(17,47)(22,26)(23,52)(33,63)(37,41)(38,56)(44,46)(48,50)(49,57)(53,58)(54,60)(55,62)(64,66)(65,67)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26520);