Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×#72209884 (order 9216) · Inn(G) = C1 · Out(G) = C2×C3×#72209884 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C22×C4 |
| abelian | C2×C42×C7 |
| multiplicative | (ℤ/435)ˣ, (ℤ/464)ˣ, (ℤ/580)ˣ, (ℤ/870)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22 | C23×C7 |
| socle, prime radical | C23×C7 | C22 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4)(11,12,13,14,15,16,17);(5,6,7,8);(9,10)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 149);