Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#21043 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C22×D4 |
| commutator | C2×C4 | C23×C4×C11 |
| Frattini, cosocle | C22×C4 | C24×C11 |
| socle | C22×C11 | C2×C4×D4 |
| prime radical | C24×C11 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C11×M16 ◁ G | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,25,30,26,29,24,38)(22,27,34,41,32,42,33,48)(23,35,36,28,37,44,39,43)(31,45,47,40,46,50,49,51);(13,16)(17,19)(20,22,25,34,26,32,24,33)(21,27,30,41,29,42,38,48)(23,31,36,47,37,46,39,49)(28,40,44,50,43,51,35,45);(12,14,15,18)(13,17,16,19)(20,23)(21,28)(22,31)(24,39)(25,36)(26,37)(27,40)(29,43)(30,44)(32,46)(33,49)(34,47)(35,38)(41,50)(42,51)(45,48);(21,29)(22,32)(28,43)(30,38)(31,46)(33,34)(35,44)(47,49)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 14031);