Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×#5934 (order 2304) · Inn(G) = C1 · Out(G) = C22×C3×#5934 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C22×C7×C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C23×C7 |
| socle, prime radical | C23×C7 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4,5,6,7,8)(13,14,15,16,17,18,19);(9,10);(11,12)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 164);