Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19071 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
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)(20,21,26,31)(22,32,35,44)(23,36,37,45)(24,30,25,29)(27,39,42,34)(28,40,43,38)(33,41,47,51)(46,48,50,49);(4,5)(6,10)(7,11)(8,16)(9,12)(13,17)(14,18)(15,19)(20,22)(21,27)(23,33)(24,39)(25,34)(26,35)(28,41)(29,44)(30,32)(31,42)(36,50)(37,47)(38,49)(40,48)(43,51)(45,46);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,23,26,37)(21,28,31,43)(22,33,35,47)(24,38,25,40)(27,41,42,51)(29,36,30,45)(32,46,44,50)(34,48,39,49);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(23,37)(28,43)(33,47)(36,45)(38,40)(41,51)(46,50)(48,49)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12059);