Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#13141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #442 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| prime radical | C23×C11 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C2×C11×#97 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,14)(16,17)(18,28)(19,31,33,39)(20,34)(21,23)(22,27)(24,29,42,47)(25,36)(26,30)(32,44,35,45)(37,43,46,41)(38,40);(12,13,15,14)(16,18,20,21,22,30,36,38)(17,23,25,28,27,40,34,26)(19,29,35,37,33,47,32,46)(24,39,43,45,42,31,41,44);(12,14,15,13)(16,19,22,33)(17,24,27,42)(18,29,30,47)(20,35,36,32)(21,37,38,46)(23,39,40,31)(25,43,34,41)(26,44,28,45)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6129);