Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = C26×AGL(1,𝔽7) (order 2688) · Inn(G) = C22×D7 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C23 |
| Fitting | C7×C22⋊C4 | C2 |
| prime radical | C23×C7 | C22 |
| hypercenter | C22⋊C4 | D7 |
| hypocenter | C7 | #126 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
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)(9,13)(12,15)(18,22)(20,23);(8,10,11,14)(9,12,13,15)(16,18,19,22)(17,20,21,23);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^5,7|a^2=e,b^2=d,c^a=cde,f^a=f^6,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6442, #90185, #90186, … |
| maximal normal subgroup of | C2×#6442, #90185, #90186, … |
| maximal quotient of | #89423, #89424, #89425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 74);