Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #6400 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#442 (order 2688) · Inn(G) = #1141 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×Q16 ⋊ C2 |
| Sylow | C7 ⋊ SD32 |
26 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)(8,9,12,15)(10,17,19,23)(11,20,18,14)(13,21,22,16);(2,3)(4,5)(6,7)(9,13)(10,16)(11,18)(14,23)(15,22)(17,20)(19,21)Polycyclic presentation:
2^5,7|b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,c^b=cd,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | D7×SD32, #89686, #89690, … |
| maximal normal subgroup of | D7×SD32, #89686, #89690, … |
| maximal quotient of | #89285, #89286, #89359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 34);