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) = C1 |
| direct product | C7×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | D4 |
| commutator | C4 | C2×C7×C8 |
| Frattini, cosocle | C2×C8 | C22×C7 |
| socle | C2×C7 | C4⋊C8 |
| prime radical | C22×C7 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C8 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,8,4,9,11,6,5,10,12,14,13,7,16,15)(17,18,19,20,21,22,23);(1,3,4,11,5,12,13,16)(2,6,7,8,10,15,9,14)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 168);