Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#2196 (order 5376) · Inn(G) = C22⋊Dic7 · Out(G) = C2×C3×D4 |
| 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)(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21);(2,3)(4,5)(6,7)(8,9,11,14,12,15,20,23)(10,16,17,22,18,13,19,21)Polycyclic presentation:
2^5,7|a^2=c,b^2=e,c^2=e,b^a=bd,d^a=de,f^a=f^6,c^b=ce41 conjugacy classes — view the full character table.
| maximal subgroup of | #89272, #89273, #89337, … |
| maximal normal subgroup of | #89272, #89273, #89337, … |
| maximal quotient of | #89323, #89325, #89338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 42);