Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = C22×#120×AGL(1,𝔽7) (order 5376) · Inn(G) = C22×D7 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, prime radical, cosocle | C22×C7 | C23 |
| Fitting | C7×C4⋊C4 | C2 |
| hypercenter | C4⋊C4 | D7 |
| hypocenter | C7 | #125 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(9,13)(10,14)(16,18,19,23)(17,20,21,22);(8,10,11,14)(9,12,13,15);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^5,7|a^2=d,b^2=e,c^2=d,c^a=cde,f^a=f^6,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #89535, #89536, #89538, … |
| maximal normal subgroup of | #89535, #89536, #89538, … |
| maximal quotient of | #89426, #89427, #89432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 85);