Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C16 · Schur multiplier (H2) = C23 |
| direct product | C3×#13554 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C16 | C23 |
| commutator | C22 | C22×C3×C16 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| socle | C22×C3 | C22×C16 |
| prime radical | C2×C3×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3)(4,5)(6,7)(8,9,11,16,12,17,24,34,13,18,25,35,27,37,46,56,14,19,26,36,28,38,47,57,29,39,48,58,49,59,64,69)(10,15,21,31,22,32,43,53,23,33,44,54,45,55,63,68,20,30,40,50,41,51,60,65,42,52,61,66,62,67,70,71);(5,7)(8,10)(9,15)(11,21)(12,22)(13,23)(14,20)(16,31)(17,32)(18,33)(19,30)(24,43)(25,44)(26,40)(27,45)(28,41)(29,42)(34,53)(35,54)(36,50)(37,55)(38,51)(39,52)(46,63)(47,60)(48,61)(49,62)(56,68)(57,65)(58,66)(59,67)(64,70)(69,71);(10,20)(15,30)(21,40)(22,41)(23,42)(31,50)(32,51)(33,52)(43,60)(44,61)(45,62)(53,65)(54,66)(55,67)(63,70)(68,71)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6542);