Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C25×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D7)⋊C8 |
| Fitting | C2×C7×C8 ⋊ C2 |
| Sylow | C7 ⋊ C22⋊C8 |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(12,13,14,16,15,17,18,19);(2,3)(4,5)(6,7)(8,9)(10,11)Polycyclic presentation:
2^5,7|b^2=c,c^2=e,b^a=bd,f^a=f^668 conjugacy classes — view the full character table.
| maximal subgroup of | #89457, C8×D28, #89460, … |
| maximal normal subgroup of | #89457, C8×D28, #89460, … |
| maximal quotient of | #89251, #89254, #89263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 26);