Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33693 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#575 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#405 ◁ G | |
512 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)(6,11,17,33)(7,20,22,55)(8,25,28,60)(9,31,29,14)(12,36,38,23)(13,40,43,30)(16,44,46,54)(18,49,26,59)(19,52,48,35)(21,42,24,58)(27,37,56,39)(32,57,64,47)(34,62,41,51)(45,67,53,65)(50,66,61,63);(4,6,10,17)(5,11,15,33)(7,16,22,46)(8,26,28,18)(9,19,29,48)(12,32,38,64)(13,41,43,34)(14,35,31,52)(20,54,55,44)(21,53,24,45)(23,57,36,47)(25,59,60,49)(27,61,56,50)(30,62,40,51)(37,66,39,63)(42,67,58,65);(4,7)(5,12)(6,16)(8,27)(9,21)(10,22)(11,32)(13,42)(14,37)(15,38)(17,46)(18,50)(19,53)(20,25)(23,30)(24,29)(26,61)(28,56)(31,39)(33,64)(34,65)(35,66)(36,40)(41,67)(43,58)(44,49)(45,48)(47,51)(52,63)(54,59)(55,60)(57,62);(6,17)(7,21,22,24)(8,28)(11,33)(12,37,38,39)(13,43)(16,45,46,53)(19,48)(20,42,55,58)(23,27,36,56)(25,60)(30,40)(32,63,64,66)(35,52)(44,67,54,65)(47,61,57,50)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26681);