Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C11 | D4 |
| commutator | C4 | C2×C42×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C23×C11 | C2×D4 |
| prime radical | C24×C11 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C11 ◁ C23×C4×C11 ◁ G | |
424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,18,19)(20,21)(22,30)(23,27)(24,28)(25,29)(26,31)(32,35)(33,34);(13,15)(20,22,24,32,25,33,23,31)(21,26,27,34,29,35,28,30);(20,23,25,24)(21,27,29,28)(22,31,33,32)(26,34,35,30)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 467);