Element structure of special linear group:SL(2,7)

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,7).
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This article gives detailed information about the element structure of special linear group:SL(2,7), which is a group of order 336.

See also element structure of special linear group of degree two.

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over $\mathbb {F} _{q}$?	Splits in $SL_{2}$ relative to $GL_{2}$?
Diagonalizable over field:F7 with distinct (and hence mutually inverse) diagonal entries	$\{2,4\}$ and $\{3,5\}$	$x^{2}-x+1$, $x^{2}+x+1$	Same as characteristic polynomial	56	2	112	Yes	Yes	No
Diagonalizable over field:F49, not over field:F7. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of $\mathbb {F} _{49}$ of norm 1	$x^{2}-3x+1$, $x^{2}-4x+1$	Same as characteristic polynomial	42	3	126	Yes	No	No
Diagonalizable over $\mathbb {F} _{q}$ with equal diagonal entries, hence a scalar	$\{1,1\}$ or $\{-1,-1\}$	$(x-a)^{2}$ where $a\in \{-1,1\}$	$x-a$ where $a\in \{-1,1\}$	1	2	2	Yes	Yes	No
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $-1$ (multiplicity 2)	$(x-a)^{2}$ where $a\in \{-1,1\}$	$x-a$ where $a\in \{-1,1\}$	24	4	96	No	No	Yes (two conjugacy classes over $GL_{2}$, each splits into two over $SL_{2}$)
Total	NA	NA	NA	NA	11	336	238	98	96