Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#13220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×#99 |
| commutator | C24 | C22×C4×C11 |
| Frattini, cosocle | C22×D4 | C23×C11 |
| prime radical | C22×D4×C11 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C2×C4×D4×C11 ◁ G | |
540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17,19,23,20,22,28,29)(18,24,26,31,25,21,27,30);(16,18,19,27)(17,21,29,31)(20,25,28,26)(22,24,23,30);(12,13,14,15)407 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6208);