I want to compute the automorphism group of a matroid. This reduces to the following problem:

Suppose I have a list of sets $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ where each $b_{ij}$ is in $\{1,\dots,n\}$. The symmetric group $S_n$ acts on this set by: $$\begin{align}&s\cdot\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\} \\ &= \{\{sb_{11},\dots,sb_{1k}\},\dots,\{sb_{k1},\dots,sb_{kk}\}\}\end{align}$$

I want to find those elements of the symmetric group on $n$ elements that stabilize the entire set $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$.

The command GroupSetwiseStabilizer seems relevant but doesn't quite do what I want.

I want to compute the automorphism group of a matroid. This reduces to the following (more general) problem:

Suppose I have a list of sets $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ where each $b_{ij}$ is in $\{1,\dots,n\}$. The symmetric group $S_n$ acts on this set by: $$\begin{align}&s\cdot\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\} \\ &= \{\{sb_{11},\dots,sb_{1k}\},\dots,\{sb_{k1},\dots,sb_{kk}\}\}\end{align}$$

I want to find those elements of the symmetric group on $n$ elements that stabilize the entire set $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$.

The command GroupSetwiseStabilizer seems relevant but doesn't quite do what I want.

I want to compute the automorphism group of a matroid. This reduces to the following problem:

Suppose I have a list of sets {{b_11,...,b_1k},...,{b_k1,...,b_kk}} where each b_ij is in {1,...,n}. The symmetric group S_n acts on this set by: s*{{b_11,...,b_1k},...,{b_k1,...,b_kk}} = {{sb_11,...,sb_1k},...,{sb_k1,...,sb_kk}}.

I want to find those elements of the symmetric group on n elements that stabilize the entire set {{b_11,...,b_1k},...,{b_k1,...,b_kk}}.

The command GroupSetwiseStabilizer seems relevant but doesn't quite do what I want.

I want to compute the automorphism group of a matroid. This reduces to the following problem:

Suppose I have a list of sets $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$ where each $b_{ij}$ is in $\{1,\dots,n\}$. The symmetric group $S_n$ acts on this set by: $$\begin{align}&s\cdot\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\} \\ &= \{\{sb_{11},\dots,sb_{1k}\},\dots,\{sb_{k1},\dots,sb_{kk}\}\}\end{align}$$

I want to find those elements of the symmetric group on $n$ elements that stabilize the entire set $\{\{b_{11},\dots,b_{1k}\},\dots,\{b_{k1},\dots,b_{kk}\}\}$.

The command GroupSetwiseStabilizer seems relevant but doesn't quite do what I want.