Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #99 |
| commutator | C2×C4 | C2×C4×C7 |
| Frattini, cosocle | C2×Q8 | C22×C7 |
| prime radical | C22×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C7×Q8 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,8,5,9,11,16)(4,7,12,6,13,14,10,15)(17,18,19,20,21,22,23);(1,3,5,11)(2,6,9,15)(4,12,13,10)(7,16,14,8)91 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 160);