Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C23×C11 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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)(18,27)(19,24)(20,33)(21,37)(22,26)(23,29)(25,35)(28,41)(30,47)(31,44)(32,42)(34,40)(36,43)(38,45)(39,46);(16,18,22,31)(17,23,26,39)(19,30,21,28)(20,34,36,45)(24,40,37,38)(25,41,42,47)(27,43,44,33)(29,32,46,35);(16,19,22,21)(17,24,26,37)(18,28,31,30)(20,32,36,35)(23,38,39,40)(25,33,42,43)(27,41,44,47)(29,45,46,34)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 873);