Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#7204 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | #99 |
| commutator | C23 | C4×C8×C11 |
| Frattini, cosocle | C2×C4×D4 | C22×C11 |
| socle | C22×C11 | #342 |
| prime radical | C22×D4×C11 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C4×C11 ◁ C2×C4×D4×C11 ◁ G | |
380 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 704 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,18,17,19,16,14)(20,21,22,25,23,24,26,27);(12,14,16,19,17,18,15,13)(21,24)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 192);