Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #120 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #379 |
| prime radical | C11×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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)(12,13,16,21,17,20,29,34)(14,22,25,35,24,18,28,31)(15,19,27,32,26,33,39,43)(23,30,38,41,37,36,40,42);(12,14)(13,18)(15,23)(16,28)(17,24)(19,30)(20,22)(21,35)(25,29)(26,37)(27,40)(31,34)(32,42)(33,36)(38,39)(41,43);(12,15)(13,19)(14,23)(16,27)(17,26)(18,30)(20,33)(21,32)(22,36)(24,37)(25,38)(28,40)(29,39)(31,41)(34,43)(35,42)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 523);