Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18765 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3482 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,12)(6,13,15,18)(7,11,8,16)(10,17,19,14)(20,21,26,31)(22,28,33,34)(23,27,32,35)(24,30,25,29);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,22,26,33)(21,27,31,35)(23,24,32,25)(28,29,34,30);(20,23)(21,28)(22,24)(25,33)(26,32)(27,29)(30,35)(31,34);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,24,26,25)(21,29,31,30)(22,32,33,23)(27,34,35,28)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11753);