Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C7×C22⋊C4 |
| unambiguous semidirect product | C22⋊(C42×C7) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C22 |
| commutator | C2 | C2×C42×C7 |
| Frattini, cosocle | C23 | C23×C7 |
| socle | C23×C7 | C23 |
| prime radical | C24×C7 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4)(13,14,15,16,17,18,19);(5,6,7,8)(9,10)(11,12);(10,12)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 785);