In 21 February 2012 three members of a dissident group called Pussy Riot, dressed in bright balaclavas, neon tights and tops, stormed the Orthodox Cathedral of Christ the Savior in Moscow and performed Mother of God, Put Putin Away. As the title suggests, it was the most defiant of anti-Kremlin protests, a sucker punch to the core of the state-church axis. 

So what prompted them to kick out against the president and his administration? Why not read the strident lyrics of their song to get a clearer picture of what it feels like to be a young woman in Putin's Russia:
 
 Virgin Mary, Mother of God, put Putin away
 Рut Putin away, put Putin away

Black robe, golden epaulettes
All parishioners crawl to bow
The phantom of liberty is in heaven
Gay-pride sent to Siberia in chains

The head of the KGB, their chief saint,
Leads protesters to prison under escort
In order not to offend His Holiness
Women must give birth and love

Shit, shit, the Lord’s shit!
Shit, shit, the Lord’s shit!

Virgin Mary, Mother of God, become a feminist
Become a feminist, become a feminist

The Church’s praise of rotten dictators
The cross-bearer procession of black limousines
A teacher-preacher will meet you at school
Go to class – bring him money!

Patriarch Gundyaev believes in Putin
Bitch, better believe in God instead
The belt of the Virgin can’t replace mass-meetings
Mary, Mother of God, is with us in protest!

Virgin Mary, Mother of God, put Putin away
 Рut Putin away, put Putin away

The act was condemned as blasphemy by the establishment and an attack on the Russian Orthodox Church, not Putin, by a cross-section of god-fearing citizens. Almost overnight, Pussy Riot had gone from being a punk group to enemies of the state, their images beamed into homes across the world. Two members, Maria 'Masha' Alyokhina and Nadezha 'Nadia' Tolokonnikova, were sentenced to two years in penal colonies for "hooliganism motivated by religious hatred". Apparently those camps – the former in Perm and the latter in Mordovia – are among the harshest in the country. Alyokhina and Tolokonnikova had petitioned to serve their sentences in Moscow, arguing that they wanted to be close to their children, both under six, but the court refused.

Meanwhile the world protested, shouting, tweeting and wearing the slogan "Free Pussy Riot". If the reaction of the Russian press was one of vilification, elsewhere the trio (third member Katja Samutsevich was later released on a suspended sentence) were celebrated as anti-authoritarian heroes. From the Arab Spring to Sao Paolo, millions of people under 30 have been in revolt, fighting for their basic human rights as individuals – particularly their freedom of expression. It quickly became clear that Pussy Riot's cause had struck a chord. As another member of the group uttered to the New Statesman during one of many illicit interviews outside of Russia: "Nobody is a prophet in their own country."

In the wake of the trial, an anti-blasphemy law is close to being approved by the Russian State Duma and it could come into effect next month if accepted by the President and the Upper House. Those who break this censorship law could face imprisonment for up to three years "for desecrating religious sites and paraphernalia". Then there are other pieces of legislation, informally known as the Pussy Riot laws, that forbid the discussion of their protests and distribution of promotional material, covering your face in public and criticising the state while speaking to a foreign journalist. (The latter calls to mind the expulsion of the Guardian's former Moscow correspondent Luke Harding, who branded Russia under Putin "a virtual mafia state". Tolerance is not the president's strong point.)

Pussy Riot are feminists, that much is clear. But are they dangerous extremists? How are their hopes and dreams any different to those of their contemporaries in the West? What do we actually know about their lives and the sacrifices they are willing to make? Thankfully we now have a few answers as the ensuing media circus and show trail has been made into a fascinating HBO documentary by Michael Lerner and Maxim Pozdorovkin, appropriately titled A Punk Prayer.

In it we are presented not only with a fly-on-the-wall account of events as they unfolded – the riot in Red Square, rehearsals, extensive courtroom footage – but we also get a chance to meet some of peripheral figures that have helped to shape the Pussy Riot story: Nadia's supportive father, her husband Piotr (a co-member of performance art group Voira and a key facilitator for this film), Masha's mother, Katja's father (who remembers his daughter devouring the work of leftist French philosophers at a prodigious age), Patriarch Kirill (a man who called Putin's rule a "miracle of god") and his "Carriers of the Cross", one of whom says this about Nadia: "The main one, she is a demon with a brain."

So much has been written and said about Pussy Riot that it can be difficult to pierce through the prevailing narrative – brash group of punks rebel in Russia. Beneath the flurry of web articles you get either comments from those who side with the group and their radical stance, or you get those knee-jerk critics who seem to miss the point.

"There music is just noise. How can you expect anyone to listen?" (It's punk, not pop.)

"Protesting in a church; they got what they deserved." (Did they really?)

Or "stupid girls, attention-seeking – get a job." (Several have one by the way.)

Their choice of protest may have been inappropriate but the attention that they have drawn to Putin's totalitarian regime and the alarmingly repressive nature of society in Russia is not. Making viewers aware of these issues is all that the directors could really have hoped to achieve over 90 minutes as they tried to establish their main characters and each woman's story. Had they not obtained the court footage we would certainly have seen a more investigative piece about a divided society subservient to its state and blinded by its faith. Part two perhaps, when Masha and Nadia are released next year.

In the trials the women come across as being very eloquent, patriotic yet progressive. They have no interest in being pop stars or celebrities. The illusion of the outspoken artist is a seductive one. It makes good headlines. But it rarely spurs people into action; the kind that draws hundreds of thousands of people out on the streets to protest your innocence. As the film progresses you quickly come to appreciate their irrepressible spirit, their conviction to stand up for what they believe in and their will to inspire others to follow suit. "Our aim is to change humanity, to free society from prejudices and stereotypes," we are told. But "only radical revolutionary action can change anything".

For co-director Lerner, the importance of a Pussy Riot in our times – in our cities – is clear. "I think we are experiencing a new wave of punk feminism around the world," he told the Guardian. "What has happened under Putin is an increased radicalisation of nationalism in Russia. It a story about the Orthodox church as an icon of nationalism. They see the group as an attack on Russia. But Pussy Riot don't want to destroy Russian society, they want to improve it. The longer they are incarerated and persecuted the stronger their message."

'Pussy Riot: A Punk Prayer' is showing at the Rich Mix, London, on Thursday 27 June at 8pm. This will be followed by a Q&A with Mike Lerner.