Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×D4⋊C8 |
| unambiguous semidirect product | D4⋊(C7×C8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D4 |
| commutator | C4 | C2×C7×C8 |
| Frattini, cosocle | C42 | C22×C7 |
| socle | C22×C7 | C22⋊C4 |
| prime radical | C2×C7×D4 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C42×C7 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,5,4,6,7,8)(9,10)(11,13)(12,15)(14,16)(17,18,19,20,21,22,23);(10,13)(11,16)(14,15)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 129);