Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C22×D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×M16 ⋊ C2 |
| Sylow | C7 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D7 |
| commutator, socle | C2×C7 | C22×C4 |
| Fitting | C7×M16 | C2 |
| prime radical | #1157 | C2 |
| cosocle | C4×C7 | C23 |
| hypercenter | M16 | D7 |
| hypocenter | C7 | #131 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ M16 | |
62 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)(8,10,11,17,12,16,18,22)(9,14,15,21,13,19,20,23);(9,13)(10,16)(15,20)(17,22);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23)Polycyclic presentation:
2^5,7|c^2=d,d^2=e,b^a=be,f^a=f^6,c^b=ce50 conjugacy classes — view the full character table.
| maximal subgroup of | #89520, #89521, #89526, … |
| maximal normal subgroup of | #89520, #89521, #89526, … |
| maximal quotient of | #89473, #89478, #89480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 102);