Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22⋊C4 |
| commutator | C22 | C2×C7×C8 |
| Frattini, cosocle | C22×C4 | C22×C7 |
| prime radical | C24×C7 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,5,4,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18,19,20,21,22,23);(9,11)(10,13)(12,16)(14,15)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 127);