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) = C22 |
No aliases for this group.
| Fitting | C7×C4⋊C4 ⋊ C2 |
| Sylow | C7 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C23 |
| Fitting | C7×C4⋊C4 | C2 |
| prime radical | C2×D28 | C2 |
| hypercenter | C4⋊C4 | D7 |
| hypocenter | C7 | #123 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(8,9,10,11)(13,17)(16,19);(9,11)(12,14,15,18)(13,16,17,19);(2,3)(4,5)(6,7)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^5,7|b^2=d,c^2=e,c^a=cd,f^a=f^6,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6457, #90198, #90205, … |
| maximal normal subgroup of | C2×#6457, #90198, #90205, … |
| maximal quotient of | #89427, #89432, #89436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 89);