What application would I need to submit to get my child's…

Your child might not need to get her permanent residence first before becoming a citizen. This will depend on how you acquired your citizenship in Canada.

Were you born in Canada ?

In that case, you are correct. I though perhaps she might be the first generation to acquire citizenship.

In order for your daughter to get her PR status in Canada, you will need to submit a sponsorship application. If you are all in Canada together, you can submit an inland sponsorship for her as your dependent child.

The average processing time for these sponsorships is approximately 12 months.

Here is the link to the application form and instruction guide from Canada Immigration's website

https://www.canada.ca/en/immigration-refugees-citizenship/services/application/application-forms-guides/guide-5289-sponsor-your-spouse-common-law-partner-conjugal-partner-dependent-child-complete-guide.html

When the processing of her application has been completed, she will be able to get a Confirmation of Landing document and then a PR card.

Hope this answers your question.

If your daughter has no permanent status in Canada, she has to be sponsored to get this status even if she is 6 years old. You cannot 'acquire' PR status as you can with citizenship. The other families you are referring to may have applied to immigrate to Canada together but if they did not, dependent children have to apply for PR through the sponsorship process.

If you want, you can call Canada Immigration's Client Support Centre to verify this. Here is the link to the contact information. It is easiest to call first thing in the morning not on a Monday for a shorter wait time.

https://www.canada.ca/en/immigration-refugees-citizenship/corporate/contact-ircc/client-support-centre.html