Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = D42×AGL(1,𝔽7) (order 2688) · Inn(G) = C2×C22⋊D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C7×Q8 ⋊ C2 |
| Sylow | C7 ⋊ #137 |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,13,20)(9,15,16,23)(10,17,12,19)(14,22,21,18);(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,18,23,22);(2,3)(4,5)(6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,17)(19,23)(20,22)Polycyclic presentation:
2^5,7|b^2=e,c^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^6,c^b=ce,d^c=de41 conjugacy classes — view the full character table.
| maximal subgroup of | #89526, #89527, #89833, … |
| maximal normal subgroup of | #89526, #89527, #89833, … |
| maximal quotient of | #89736, #89738, #89739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 137);