Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1266 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)(4,5,11,17)(6,18,23,34)(7,24,27,40)(8,14,29,39)(9,16,10,15)(12,22,35,21)(13,30,37,26)(19,44,47,32)(20,43,49,60)(25,53,56,52)(28,41,31,38)(33,51,58,48)(36,54,61,55)(42,50,59,46)(45,57,66,63)(62,64,67,65);(4,6,10,22,11,23,9,21)(5,12,16,34,17,35,15,18)(7,19,30,50,27,47,26,46)(8,20,31,51,29,49,28,48)(13,32,40,59,37,44,24,42)(14,33,41,60,39,58,38,43)(25,45,55,65,56,66,54,64)(36,57,53,67,61,63,52,62);(4,7)(5,13)(6,19)(8,25)(9,26)(10,30)(11,27)(12,32)(14,36)(15,24)(16,40)(17,37)(18,42)(20,45)(21,46)(22,50)(23,47)(28,54)(29,56)(31,55)(33,57)(34,59)(35,44)(38,52)(39,61)(41,53)(43,62)(48,64)(49,66)(51,65)(58,63)(60,67);(4,8)(5,14)(6,20)(7,25)(9,28)(10,31)(11,29)(12,33)(13,36)(15,38)(16,41)(17,39)(18,43)(19,45)(21,48)(22,51)(23,49)(24,52)(26,54)(27,56)(30,55)(32,57)(34,60)(35,58)(37,61)(40,53)(42,62)(44,63)(46,64)(47,66)(50,65)(59,67)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25940);