Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = C22×S4×AGL(1,𝔽7) (order 4032) · Inn(G) = C22×D7 · Out(G) = C22×C3×D3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D7 |
| commutator, socle | C2×C7 | C22×C4 |
| Fitting, prime radical | C7×#41 | C2 |
| cosocle | C4×C7 | C23 |
| hypercenter | #41 | D7 |
| hypocenter | C7 | #131 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ #41 | |
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)(10,16)(13,19)(17,22)(20,23);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23);(2,3)(4,5)(6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)Polycyclic presentation:
2^5,7|a^2=d,d^2=e,f^a=f^6,c^b=ce50 conjugacy classes — view the full character table.
| maximal subgroup of | #89595, #89596, #89968, … |
| maximal normal subgroup of | #89595, #89596, #89968, … |
| maximal quotient of | #89769, #89771, #89772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 143);