Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #117 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #382 |
| prime radical | C23×C11 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C11×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,22,25,18)(15,26,28,41)(16,21)(17,20)(19,27,32,42)(23,36)(24,35,39,31)(29,33)(30,37)(34,43)(38,40);(12,14,16,24,17,25,29,39)(13,18,21,31,20,22,33,35)(15,23,27,37,28,38,42,43)(19,30,26,40,32,34,41,36);(12,15)(13,19,20,32)(14,23)(16,27)(17,28)(18,30,22,34)(21,26,33,41)(24,37)(25,38)(29,42)(31,40,35,36)(39,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 565);