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Sterling steadies near one-week high above $1.30

* Graphic: sterling and gilt yields http://bit.ly/2dgAXn1

* Graphic: World FX rates in 2017 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv

LONDON, July 25 (Reuters) - Sterling steadied close to a one-week high just above $1.30 on Tuesday, with hopes for increasing political stability and an upbeat business survey lending support to the currency.

Data showed British factories increased output at the fastest rate since the mid-1990s over the past three months, suggesting manufacturing might help to support the economy as it slows during 2017.

Sterling hit as high as $1.3084 after the data, before easing back to trade slightly up on the day at $1.3033 by 1630 GMT. That was less than a cent away from last week's high of $1.3126, the strongest since September.

The pound recovered from its worst week against the euro in nine months on Monday, as investors took profit on gains for the single currency and after ministers talked up a transitional deal to smooth Britain's exit from the European Union.

British trade minister Liam Fox said on Sunday that he backed such a deal, but that it would have to come to an end before the next election due in 2022.

"Were more optimistic the recent more unified approach in the Cabinet towards aiming to secure a longer-term transitional deal is in our view a positive for the pound," said MUFG currency economist Lee Hardman. "But the market isn't fully ready to buy into that view at this stage."

"Brexit developments are still by far the most important driver of pound performance."

A cut in the IMF's growth forecast for this year for the world's fifth biggest economy on Monday underlined the risk that Britain will struggle in the years ahead as a housing price boom stalls and companies shift operations to mainland Europe.

"Overall the Cable (dollar) rate has struggled to sustain rallies above $1.3000, weighed down by recent softer UK inflation readings and ongoing worry over the outlook," said LMAX Exchange analyst Joel Kruger.

"Still, while broad based dollar-selling persists, any setbacks in the UK currency should be able to find decent support."

Traders will be looking out for Wednesday's British GDP data, which economists expect will show 1.7 percent year-on-year growth. (Editing by Mark Potter and Pritha Sarkar)