Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1388 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #102 |
| commutator | C8 | C42×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C2×C11 | #340 |
| prime radical | C23×C11 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,22,19,24,18,35)(14,25,27,37,30,39,29,43)(15,31,33,40,34,23,17,21)(20,28,36,32,38,42,41,26);(12,14,17,28,18,29,34,26,19,30,33,42,16,27,15,32)(13,20,23,37,22,36,21,39,24,38,31,43,35,41,40,25)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 120);