Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#23900 (order 10752) · Inn(G) = C2×D7 · Out(G) = C3×#2403 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C7 ⋊ #117 |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(2,3)(4,5)(6,7)(8,9,10,11)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^5,7|a^2=d,b^2=e,c^a=ce,f^a=f^668 conjugacy classes — view the full character table.
| maximal subgroup of | #89318, #89326, #89350, … |
| maximal normal subgroup of | #89318, #89326, #89350, … |
| maximal quotient of | #89697, #89698, #89704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 121);