Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1132 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#3003 (order 5376) · Inn(G) = D28 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #106 |
36 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)(12,13,14,16,15,17,18,19);(2,3)(4,5)(6,7)(8,9,10,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,7|a^2=c,b^2=d,d^2=e,b^a=bd,d^a=de,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | #89289, #89308, #89313, … |
| maximal normal subgroup of | #89289, #89308, #89313, … |
| maximal quotient of | #89252, #89308, #89346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 23);