Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#33441 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | C2×D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×C8×D4 |
| prime radical | C3×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C22×C3×C16 ◁ G | |
646 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)(4,5,9,16,10,15,29,40,11,14,30,39,8,26,28,52)(6,17,20,42,19,12,31,34,21,35,46,59,27,33,53,58)(7,13,23,38,24,37,50,61,25,36,51,60,22,47,49,64)(18,41,44,62,43,32,54,56,45,57,63,67,48,55,65,66);(4,6)(5,12)(7,18)(8,27)(9,20)(10,19)(11,21)(13,32)(14,33)(15,35)(16,34)(17,26)(22,48)(23,44)(24,43)(25,45)(28,53)(29,31)(30,46)(36,55)(37,57)(38,56)(39,58)(40,59)(41,47)(42,52)(49,65)(50,54)(51,63)(60,66)(61,67)(62,64);(4,7,9,23,10,24,29,50,11,25,30,51,8,22,28,49)(5,13,16,38,15,37,40,61,14,36,39,60,26,47,52,64)(6,18,20,44,19,43,31,54,21,45,46,63,27,48,53,65)(12,32,34,56,35,57,59,67,33,55,58,66,17,41,42,62);(5,14)(12,33)(13,36)(15,26)(16,39)(17,35)(32,55)(34,58)(37,47)(38,60)(40,52)(41,57)(42,59)(56,66)(61,64)(62,67)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26429);