Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#13562 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C11 |
| Frattini, cosocle | #127 | C23×C11 |
| prime radical | C11×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C22×C4×C11 ◁ C11×#532 ◁ G | |
490 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19)(14,17,22,18)(16,20,25,26)(21,24,27,23);(12,13)(14,21)(15,18)(16,20)(17,23)(19,22)(24,26)(25,27);(12,14,16,23)(13,17,20,21)(15,22,25,24)(18,26,27,19)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6550);