Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×#41×AGL(1,𝔽7) (order 1344) · Inn(G) = D7 · Out(G) = C2×C3×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C7⋊C16 |
| unambiguous semidirect product | (C2×C7)⋊C16 |
| derived, Sylow | C7 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D7 |
| commutator, hypocenter | C7 | C2×C16 |
| Frattini | C8 | C2×D7 |
| Fitting | C2×C7×C8 | C2 |
| socle, prime radical | C22×C7 | C8 |
| cosocle | C7×C8 | C22 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C8 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2)(4,5)(6,7)(8,9)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(1,2)(3,4,6,8,9,7,5)80 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C7⋊C16, #89256, C4⋊(C7⋊C16), … |
| maximal normal subgroup of | C4×C7⋊C16, #89256, C4⋊(C7⋊C16), … |
| maximal quotient of | C4×C7⋊C16, C4⋊(C7⋊C16), C2×C7⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 17);