Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1379 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | D4 |
| commutator | C4 | C4×C8×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C22×C11 | C22⋊C8 |
| prime radical | C23×C11 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ C22×C8×C11 ◁ G | |
136 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 88 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)(16,17,20,25,21,26,35,43,22,27,36,44,19,24,33,41)(18,28,30,45,29,39,46,42,31,40,34,47,32,23,37,38);(12,13,14,15)(16,18)(17,23)(19,32)(20,34)(21,29)(22,31)(24,40)(25,42)(26,28)(27,39)(30,36)(33,46)(35,37)(38,44)(41,45)(43,47)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 111);