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.

Conjugacy class structure

Compare with element structure of special linear group of degree two#Conjugacy class structure.

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 $a\in \{-1,1\}$	Same as characteristic polynomial	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	240	114	96