Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C2×C3×#2196 (order 768) · Inn(G) = C22⋊C4 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22⋊C4 |
| commutator | C22 | C2×C4×C7 |
| Frattini, cosocle | C2×C4 | C22×C7 |
| prime radical | C2×C7×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3,6,4,5,7,8)(9,10,11,12,13,14,15);(2,5)(3,7)77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 49);