Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | D8 |
| commutator | C8 | C42×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C22×C11 | #102 |
| prime radical | C23×C11 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15)(16,17)(18,27)(19,32)(20,35)(21,37)(22,26)(23,28)(24,34)(25,33)(29,39)(30,47)(31,43)(36,40)(38,46)(41,45)(42,44);(16,18,20,29,33,44,34,45,22,31,19,28,21,30,36,46)(17,23,24,27,37,41,35,47,26,39,40,43,25,38,32,42)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 114);