Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
924 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)(6,13)(7,11)(8,16)(9,12)(10,14)(15,18)(17,19)(20,21,25,29)(22,30,33,34)(23,28,24,27)(26,31,35,32);(4,6,9,15)(5,10,12,19)(7,14,8,17)(11,18,16,13)(20,22,25,33)(21,26,29,35)(23,31,24,32)(27,34,28,30);(4,5)(6,10)(7,16)(8,11)(9,12)(13,14)(15,19)(17,18)(21,27)(23,24)(26,34)(28,29)(30,35)(31,32);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,23,25,24)(21,28,29,27)(22,31,33,32)(26,30,35,34)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11680);