Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1423 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | D16 |
| commutator | C16 | C2×C4×C11 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 2 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15)(17,22)(18,23)(19,27)(20,30)(24,39)(25,42)(26,35)(28,44)(29,38)(31,43)(32,33)(34,40)(36,37)(41,47)(45,46);(16,17,23,33,19,24,36,42,30,40,44,47,43,45,29,35,21,26,38,46,31,41,28,34,20,25,37,39,27,32,18,22)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 155);