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) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #103 |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,6,7,5,3)(8,10,11,9)(12,13,14,16,15,17,18,19);(2,3)(4,5)(6,7)(8,9,11,10)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,7|a^2=d,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 | #89453, C4×#1132, #89469, … |
| maximal normal subgroup of | #89453, C4×#1132, #89469, … |
| maximal quotient of | #89251, #89262, #89276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 22);