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.
| Sylow | C7 ⋊ #101 |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,11,17,12,18,20,23)(9,13,14,21,15,16,19,22);(2,3)(4,5)(6,7)(8,9,12,15)(10,16,18,13)(11,19,20,14)(17,21,23,22)Polycyclic presentation:
2^5,7|a^2=e,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 | #89517, #89520, D7×#101, … |
| maximal normal subgroup of | #89517, #89520, D7×#101, … |
| maximal quotient of | #89260, #89264, #89277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 30);