Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
868 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)(6,13)(7,11)(8,16)(9,12)(10,14)(15,18)(17,19)(20,21,25,29)(22,26,32,39)(23,27,34,41)(24,35,36,28)(30,42,44,37)(31,45,46,38)(33,47,48,40)(43,49,51,50);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,22)(21,26)(23,30)(24,31)(25,32)(27,37)(28,38)(29,39)(33,43)(34,44)(35,45)(36,46)(40,49)(41,42)(47,50)(48,51);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,23)(21,27)(22,30)(24,33)(25,34)(26,37)(28,40)(29,41)(31,43)(32,44)(35,47)(36,48)(38,49)(39,42)(45,50)(46,51);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,24,25,36)(21,28,29,35)(22,31,32,46)(23,33,34,48)(26,38,39,45)(27,40,41,47)(30,43,44,51)(37,49,42,50)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11303);