Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
838 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)(6,13,16,19)(7,11,8,17)(10,14,18,15)(20,21)(22,29)(23,27)(24,32)(25,28)(26,30)(31,34)(33,35);(4,6,9,16)(5,10,12,18)(7,15,8,14)(11,19,17,13)(20,21)(22,26)(23,27)(24,32)(25,28)(29,33)(30,34)(31,35);(4,5)(6,10)(7,11)(8,17)(9,12)(13,14)(15,19)(16,18)(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,13,18,19)(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12029);