Sporadic groups

[Velvetski]

I'd like to ask if I'm allowed to make a group out of all the sporadic groups by directly multiplying them (just muliplying the orders), with the resulting group having an order of about 2.333×10^365?
Also, is it possible to calculate something like a geometric mean between a sporadic group (i.e. 7920) and the resulting group (order of 2.333×10^365)? Thank you!

 

[blamocur]

I'd like to ask if I'm allowed to make a group out of all the sporadic groups by directly multiplying them (just muliplying the orders), with the resulting group having an order of about 2.333×10^365?

I don't see what would keep you from doing this. But good luck building the multiplication table

Also, is it possible to calculate something like a geometric mean between a sporadic group (i.e. 7920) and the resulting group (order of 2.333×10^365)? Thank you!

Are you asking how to calculate a geometric mean of the orders of the two groups?

 

[Velvetski]

OK. so I take it as yes (that the group of order ~ 2.333×10^365) is a legal one.

Well, I'd like to take Mathieu group M11 (order 7920) and the resulting group (order of ~ 2.333×10^365) and get the number 4.3×10^184 out of it. Treating it as a simple geometric mean of two numbers. Would that be a legal step?
Thank you very much!

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[blamocur]

OK. so I take it as yes (that the group of order ~ 2.333×10^365) is a legal one.

I am not an expert on group theory, but I don't see anything illegal in taking direct products of any number of arbitrary groups. And, AFAIK, the order of the product is always the product of orders.

Well, I'd like to take Mathieu group M11 (order 7920) and the resulting group (order of ~ 2.333×10^365) and get the number 4.3×10^184 out of it. Treating it as a simple geometric mean of two numbers. Would that be a legal step?
Thank you very much!

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Get the number out of two groups? I am suspecting you are trying to get not simply a number but a new group whose order is the geometric mean of the orders of the two groups you mentioned. If that's your question then sorry, I have no clue how to do that or whether it is doable at all.