Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#1373 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | D8 |
| commutator | C8 | C2×C8×C11 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| socle | C2×C11 | #335 |
| prime radical | C22×C11 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,21,17,23,15,31,18,24,26,20,32,22,33,38)(14,25,28,35,29,40,27,19,30,34,41,36,42,39,43,37);(12,14,26,41,17,29,33,43,18,30,16,28,32,42,15,27)(13,19,31,39,22,35,21,34,24,37,38,40,23,36,20,25)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 105);