Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#13195 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | C2×D4 |
| commutator | C2×C4 | C22×C8×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C22×C11 | C8×D4 |
| prime radical | C2×C11×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ C22×C11×C16 ◁ G | |
438 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)(13,15)(16,17,22,28,23,27,37,43,20,26,30,25,21,38,19,36)(18,29,34,40,35,44,46,47,32,24,39,42,33,41,31,45);(12,13,14,15)(16,18,30,39,23,35,19,31,20,32,22,34,21,33,37,46)(17,24,25,40,27,41,36,47,26,29,28,42,38,44,43,45);(16,19,21,30,20,37,23,22)(17,25,27,36,26,28,38,43)(18,31,33,39,32,46,35,34)(24,40,41,47,29,42,44,45)968 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6183);