Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = D42×AGL(1,𝔽7) (order 2688) · Inn(G) = (C2×D7)⋊C4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×M16 ⋊ C2 |
| Sylow | C7 ⋊ #100 |
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:
(1,2,4,6,7,5,3)(8,9,10,13,11,12,14,15);(2,3)(4,5)(6,7)(9,12)(10,14)Polycyclic presentation:
2^5,7|b^2=c,c^2=e,b^a=bd,c^a=ce,f^a=f^6,d^b=de41 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#100, #89521, #89523, … |
| maximal normal subgroup of | D7×#100, #89521, #89523, … |
| maximal quotient of | #89259, #89263, #89276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 29);