Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#13135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #459 |
| 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×C4×C8×C11 ◁ G | |
324 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)(12,13,15,14)(16,17)(18,26)(19,31)(20,36)(21,38)(22,27)(23,29)(24,33)(25,37)(28,42)(30,44)(32,46)(34,43)(35,47)(39,41)(40,45);(12,14,15,13)(16,18,20,29,22,30,37,21)(17,23,25,26,27,38,36,44)(19,28,32,45,35,33,34,39)(24,31,40,43,42,47,41,46);(16,19,20,32,22,35,37,34)(17,24,25,40,27,42,36,41)(18,28,29,45,30,33,21,39)(23,31,26,43,38,47,44,46)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6123);