Element structure of special linear group:SL(2,9)

This article describes the element structure of special linear group:SL(2,9).

Order computation
The group $$SL(2,9)$$ has order 720. with prime factorization $$720 = 2^4 \cdot 3^2 \cdot 5^1 = 16 \cdot 9 \cdot 5$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 720:

Interpretation as double cover of alternating group
$$SL(2,9)$$ is isomorphic to $$2 \cdot A_n,n = 6$$. Recall that we have the following rules to determine splitting and orders. The rules listed below are only for partitions that already correspond to even permutations, i.e., partitions that have an even number of even parts:

Number of conjugacy classes
The group has 13 conjugacy classes. The number can be computed in a number of ways:

{| class="sortable" border="1" ! Family !! Parameter values !! Formula for number of conjugacy classes of a group in the family !! Proof or justification of formula !! Evaluation at parameter values !! Full interpretation of conjugacy class structure
 * special linear group of degree two $$SL(2,q)$$ over a finite field of size $$q$$ || $$q = 9$$, i.e., field:F9 || Case $$q$$ odd: $$q + 4$$ Case $$q$$ even: $$q + 1$$ || element structure of special linear group of degree two over a finite field; see also [[number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size || Since 9 is odd, we use the odd case formula, and get $$q + 4 = 9 + 4 = 9$$ ||
 * double cover of alternating group $$2 \cdot A_n$$ || $$n = 6$$, i.e., the group is double cover of alternating group:A6 || (number of unordered integer partitions of $$n$$) + 3(number of partitions of $$n$$ into distinct odd parts) - (number of partitions of $$n$$ with a positive even number of even parts and with at least one repeated part) || See element structure of double cover of alternating group, splitting criterion for conjugacy classes in double cover of alternating group || For $$n = 6$$, the three numbers to calculate are respectively 11,1,1. So, we get $$11 + 3(1) - 1 = 13$$. ||
 * double cover of alternating group $$2 \cdot A_n$$ || $$n = 6$$, i.e., the group is double cover of alternating group:A6 || (number of unordered integer partitions of $$n$$) + 3(number of partitions of $$n$$ into distinct odd parts) - (number of partitions of $$n$$ with a positive even number of even parts and with at least one repeated part) || See element structure of double cover of alternating group, splitting criterion for conjugacy classes in double cover of alternating group || For $$n = 6$$, the three numbers to calculate are respectively 11,1,1. So, we get $$11 + 3(1) - 1 = 13$$. ||
 * double cover of alternating group $$2 \cdot A_n$$ || $$n = 6$$, i.e., the group is double cover of alternating group:A6 || (number of unordered integer partitions of $$n$$) + 3(number of partitions of $$n$$ into distinct odd parts) - (number of partitions of $$n$$ with a positive even number of even parts and with at least one repeated part) || See element structure of double cover of alternating group, splitting criterion for conjugacy classes in double cover of alternating group || For $$n = 6$$, the three numbers to calculate are respectively 11,1,1. So, we get $$11 + 3(1) - 1 = 13$$. ||