Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C11×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C42×C11 ◁ G | |
218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 218 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) = 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)(13,18,20,32)(14,23)(15,28)(19,34,35,40)(21,33,22,29)(24,39,36,31)(26,42)(27,38)(30,41)(37,43);(12,13)(14,24)(15,29)(16,21)(17,22)(18,30)(19,27)(20,25)(23,35)(26,39)(28,32)(31,42)(33,41)(34,43)(36,38)(37,40);(12,14,25,38)(13,19,20,35)(15,26,41,42)(16,27,17,23)(18,31,32,39)(21,24,22,36)(28,43,30,37)(29,34,33,40)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 253);