Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×#120×AGL(1,𝔽7) (order 2688) · Inn(G) = D7 · Out(G) = C2×C3×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C8×D7 |
| Fitting | C2×C7×C8 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D7 |
| commutator, hypocenter | C7 | C22×C8 |
| Frattini | C4 | C22×D7 |
| Fitting | C2×C7×C8 | C2 |
| socle | C22×C7 | C2×C4 |
| prime radical | C22×D7 | C4 |
| cosocle | C4×C7 | C23 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C8 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(4,5)(6,7)(8,9);(3,4,6,8,9,7,5)(10,11,12,14,13,15,16,17);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊C16, C4×C8×D7, C8×D28, … |
| maximal normal subgroup of | (C2×D7)⋊C16, C4×C8×D7, C8×D28, … |
| maximal quotient of | C8×Dic14, C4×C8×D7, #89457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 94);