Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22⋊C4 |
| commutator | C22 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C2×C7 | C2×C22⋊C4 |
| prime radical | C2×C7×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,7,5,8,13,16)(3,9,11,14,12,6,10,15)(17,18,19,20,21,22,23);(1,3)(2,6)(4,10)(5,12)(7,14)(8,9)(11,13)(15,16);(1,4,5,13)(2,7,8,16)(3,10,12,11)(6,14,9,15)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 821);