Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C2×C11×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
220 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,22,26)(18,27,30,45)(19,24,21,36)(20,32,35,46)(23,38,41,29)(25,31,43,34)(28,44,37,42)(33,39,47,40);(16,18)(17,23)(19,28)(20,33)(21,37)(22,30)(24,39)(25,42)(26,41)(27,32)(29,34)(31,38)(35,47)(36,40)(43,44)(45,46);(13,15)(16,19,22,21)(17,24,26,36)(18,28,30,37)(20,31,35,34)(23,39,41,40)(25,32,43,46)(27,44,45,42)(29,33,38,47)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 921);