Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#12687 (order 10752) · Inn(G) = C22×D7 · Out(G) = C3×#379 |
| 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 | C42×C7 | C2 |
| hypercenter | C42 | D7 |
| hypocenter | C7 | #125 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
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)(12,15)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,12,13,15)(17,20)(21,23);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,21,22,23)Polycyclic presentation:
2^5,7|a^2=e,b^2=d,c^2=e,b^a=be,c^a=cd,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | #89260, #89455, #89469, … |
| maximal normal subgroup of | #89260, #89455, #89469, … |
| maximal quotient of | #89429, #89432, #89700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 65);