Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#102 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1102 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)(4,5,9,14,10,16,30,15)(6,17,20,39,21,44,29,43)(7,12,24,37,22,35,51,38)(8,25,27,13,28,40,56,41)(11,26,32,19,34,48,33,49)(18,42,47,61,45,63,55,64)(23,52,54,36,50,62,67,60)(31,53,58,46,57,66,59,65);(4,6,10,21)(5,11,16,34)(7,18,22,45)(8,26,28,48)(9,29,30,20)(12,31,35,57)(13,39,41,43)(14,33,15,32)(17,25,44,40)(19,27,49,56)(23,53,50,66)(24,55,51,47)(36,61,60,64)(37,59,38,58)(42,52,63,62)(46,54,65,67);(4,7)(5,12)(6,18)(8,23)(9,24)(10,22)(11,31)(13,36)(14,37)(15,38)(16,35)(17,42)(19,46)(20,47)(21,45)(25,52)(26,53)(27,54)(28,50)(29,55)(30,51)(32,58)(33,59)(34,57)(39,61)(40,62)(41,60)(43,64)(44,63)(48,66)(49,65)(56,67);(7,22)(12,35)(18,45)(23,50)(24,51)(31,57)(36,60)(37,38)(42,63)(46,65)(47,55)(52,62)(53,66)(54,67)(58,59)(61,64)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26381);