Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1132 |
| automorphism | Aut(G) = C23×D4×AGL(1,𝔽7) (order 2688) · Inn(G) = D28 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C7×C8 ⋊ C2 |
| Sylow | C7 ⋊ #102 |
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)(8,9,10,11)(12,13,14,16,15,17,18,19);(2,3)(4,5)(6,7)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,7|b^2=cd,c^2=e,b^a=bc,c^a=ce,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1132, C4×D56, #89466, … |
| maximal normal subgroup of | C4×#1132, C4×D56, #89466, … |
| maximal quotient of | #89254, #89265, #89280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 27);