Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C42 |
| commutator | C4 | C2×C42×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C2×#97 |
| prime radical | C23×C11 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ G | |
160 conjugacy classes of subgroups.
| 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)(12,13,15,19,16,21,27,35,17,20,28,34,29,36,42,43)(14,22,24,38,25,31,40,33,23,37,26,41,39,18,30,32);(12,14)(13,18,20,31)(15,26)(16,25)(17,23)(19,33,34,32)(21,22,36,37)(24,28)(27,30)(29,39)(35,41,43,38)(40,42);(14,23)(18,31)(22,37)(24,26)(25,39)(30,40)(32,33)(38,41)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 842);