Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,15)(8,10)(9,14)(11,16)(13,19)(17,18)(20,21,26,31)(22,32,35,42)(23,36,38,44)(24,29,25,30)(27,39,43,34)(28,40,45,37)(33,41,49,51)(46,47,50,48);(7,16)(11,15)(12,19)(14,17)(20,22,26,35)(21,27,31,43)(23,33,38,49)(24,39,25,34)(28,41,45,51)(29,32,30,42)(36,50,44,46)(37,48,40,47);(4,6)(5,9)(7,17)(8,13)(10,18)(11,19)(12,15)(14,16)(20,23)(21,28)(22,33)(24,37)(25,40)(26,38)(27,41)(29,44)(30,36)(31,45)(32,46)(34,47)(35,49)(39,48)(42,50)(43,51);(5,10)(9,18)(11,15)(12,19)(20,24,26,25)(21,29,31,30)(22,34,35,39)(23,37,38,40)(27,42,43,32)(28,44,45,36)(33,47,49,48)(41,50,51,46)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12079);