Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×B(4,𝔽2) (order 2688) · Inn(G) = C22⋊Dic7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #99 |
52 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)(8,10)(9,12)(11,15)(13,14);(2,3)(4,5)(6,7)(8,9)(10,14,15,12)(11,13)Polycyclic presentation:
2^5,7|a^2=c,b^a=bd,d^a=de,f^a=f^6,c^b=ce41 conjugacy classes — view the full character table.
| maximal subgroup of | #89268, #89269, #89270, … |
| maximal normal subgroup of | #89268, #89269, #89270, … |
| maximal quotient of | #89320, #89321, #89323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 40);