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Save the Children led international condemnation of the blast targeting a defence ministry building, which sent a plume of smoke into the air during rush hour and shook buildings nearly two kilometres (1.2 miles) away.
Gunmen then stormed a nearby building, triggering a gun battle with special forces in the Puli Mahmood Khan neighbourhood of the Afghan capital.
At least six people were killed, including one child and two special forces, said interior ministry spokesman Nasrat Rahimi.
The health ministry put the wounded toll at 116 people. Among the injured were 50 children, officials said, adding that most had been hurt by flying glass and were in stable condition.
Rahimi said all five gunmen and the driver of the car had been killed, and a clearing operation was over.
Some social media images purportedly taken at a hospital showed wounded, stunned children in school uniforms, still clutching books as they arrived for treatment.
Five schools were damaged in the blast, the education ministry said.
Save the Children branded the attack "utterly deplorable", warning that "children´s smaller bodies sustain more serious injuries than adults" and that the trauma of such attacks can stay with them for years.
'The world turned upside down'
The Taliban claimed the attack, which came just two days after the insurgents began a seventh round of talks with the US in Qatar, as Washington eyes a breakthrough before Afghanistan´s presidential election in September.
"We were sitting inside the office when the world turned upside down on us," said Zaher Usman, an employee at a branch of the culture ministry, which he said stands just 150 metres (yards) from the blast site.
"When I opened my eyes, the office was filled with smoke and dust and everything was broken, my colleagues were screaming," Usman told AFP by telephone.
AFP reporters could hear gunshots and multiple smaller explosions for hours after the initial blast, before the clearing operation was announced.
The interior ministry said 210 civilians had been rescued from buildings nearby.
Taliban reject talks with Kabul
Nearby Shamshad TV station, which was attacked in 2017, aired images of broken glass and damage to its offices.
"I was terrified," Shamshad anchor Hashmat Stanikzai told AFP.
A media watchdog said seven Shamshad journalists were among the wounded.
The attack came as the US was set to resume negotiations with the militants in Doha.
With the attack still ongoing, the Taliban spokesman in Doha again insisted that the insurgents will not negotiate with Kabul.
"Once the timeline for the withdrawal of foreign forces is set in the presence of international observers, then we will begin the talks to the Afghan side, but we will not talk to the Kabul administration as a government," Suhail Shaheen tweeted.
There are deep concerns among Afghans who fear Washington will rush for the exits and allow the militants to return to some semblance of power.
However, US officials have insisted that "nothing is agreed until everything is agreed", including intra-Afghan talks.
