Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = (C2×D7)⋊C4 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×C22⋊C4 ⋊ C2 |
| Sylow | C7 ⋊ #99 |
52 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)(8,10)(9,12,13,14)(11,15);(2,3)(4,5)(6,7)(8,9)(10,14)(11,13)(12,15)Polycyclic presentation:
2^5,7|b^2=c,b^a=bd,c^a=ce,f^a=f^6,d^b=de41 conjugacy classes — view the full character table.
| maximal subgroup of | #89512, #89513, #89514, … |
| maximal normal subgroup of | #89512, #89513, #89514, … |
| maximal quotient of | #89246, #89247, #89262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 12);