Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1132 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = D28 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C42×C7 ⋊ C2 |
| Sylow | C7 ⋊ #104 |
44 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)(9,12,13,14);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^5,7|b^2=c,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | #89472, #89489, D7×#104, … |
| maximal normal subgroup of | #89472, #89489, D7×#104, … |
| maximal quotient of | #89246, #89247, #89251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 11);