# Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$?

This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; $S_3$ acts on the elementary abelian $2$-subgroup by permuting three copies of $\mathbb{Z}_2$.

(i.e. if $\langle a_1\rangle \times \langle a_2\rangle \times \langle a_3\rangle$ is the elementary abelian $2$-group, and $\sigma\in S_3$, the action of $\sigma$ on the elementary abelian $2$-group is $\sigma (a_1^{\epsilon_1} a_2^{\epsilon_2}a_3^{\epsilon_3})\sigma^{-1}=a_{\sigma(1)}^{\epsilon_1} a_{\sigma(2)}^{\epsilon_2}a_{\sigma(3)}^{\epsilon_3}$ and $\epsilon_i\in\{0,1\}$.

[Unfortunately, I don't have GAP.]

• Shall the order be 24 and not 48? Jan 2, 2015 at 20:53
• Oh! yes! I am very sorry! I wanted $S_3$. Thanks "Alexander" for suggestion! Jan 4, 2015 at 4:08
• I think the new notation is still confusing - what you really want is a semidirect product of an elementary abelian group of order $8$ and $S_3$ ? Jan 4, 2015 at 10:11
• The group you describe is the wreath product $C_2\wr S_3$ (for the natural action of $S_3$. It has 98 subgroups in 33 conjugacy classes and listing them (or even drawing the subgroup lattice) gets messy very soon. What Information are you actually looking for? Jan 5, 2015 at 16:57
• OK! Does there exists subgroups of order $24$? What are they? Jan 6, 2015 at 8:33

1. $ℤ_2^3 ⋊ A_3$
2. A subgroup, isomorphic to $S_4$ generated by $a_1a_2, a_2a_3, \sigma, \tau$, where $\sigma$ (order 3) and $\tau$ (order 2) are the generators of the original $S_3$ complement. It corresponds to the permutation action of $S_4$ on the cosets of $\langle(1,2),(3,4)\rangle$ (intransitive $C_2\times C_2$).
3. A subgroup, also isomorphic to $S_4$, generated by $a_1a_2, a_2a_3, \sigma, a_1\cdot\tau$. It corresponds to the permutation action of $S_4$ on the cosets of $\langle(1,2,3,4)\rangle$ (subgroup $C_4$).
In the classification of transitive groups of degree 3 (your group is a wreath product $C_2\wr S_3$) from http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6560180&fileId=S1461157000000115