Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#1740 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C22⋊C4 |
| commutator | C22 | C2×C42×C11 |
| Frattini, cosocle | C24 | C23×C11 |
| socle | C22×C11 | C2×C22⋊C4 |
| prime radical | C24×C11 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C23×C4×C11 ◁ G | |
420 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17)(18,24,26,21)(19,22)(20,23)(25,30,31,28)(27,29);(12,13,14,15)(16,18)(17,21)(19,25)(20,26)(22,28)(23,24)(27,31)(29,30);(16,19,20,27)(17,22,23,29)(18,25,26,31)(21,28,24,30)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 472);