Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1321 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #99 |
| commutator | C2×C4 | C2×C8×C11 |
| Frattini, cosocle | C2×M16 | C22×C11 |
| socle | C2×C11 | #323 |
| prime radical | C2×C11×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×M16 ◁ G | |
112 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)(12,13,16,21,17,20,31,37,18,22,32,39,15,27,29,38)(14,23,25,40,24,19,33,34,26,36,30,42,28,35,41,43);(12,14)(13,19)(15,28)(16,30)(17,24)(18,26)(20,36)(21,34)(22,35)(23,27)(25,32)(29,33)(31,41)(37,42)(38,40)(39,43)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 53);