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×#19040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
872 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)(5,10)(6,13)(7,8)(12,15)(14,16)(17,19)(20,21)(22,29)(23,27)(24,33)(25,28)(26,31)(30,34)(32,35);(4,5,9,12)(6,11,14,18)(7,15,8,10)(13,19,16,17)(20,22,25,32)(21,26,28,34)(23,31,24,30)(27,35,33,29);(4,6)(5,11)(7,13)(8,16)(9,14)(10,17)(12,18)(15,19)(20,21)(22,26)(23,27)(24,33)(25,28)(29,30)(31,35)(32,34);(5,12)(10,15)(11,18)(17,19)(20,23,25,24)(21,27,28,33)(22,30,32,31)(26,29,34,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12028);