Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C23 |
| commutator | C22 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C22×C7 | C22×C4 |
| prime radical | C23×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,14,12,13,15,16)(17,18,19,20,21,22,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,13)(14,16)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 840);