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.]

  • $\begingroup$ Shall the order be 24 and not 48? $\endgroup$ Jan 2, 2015 at 20:53
  • $\begingroup$ Oh! yes! I am very sorry! I wanted $S_3$. Thanks "Alexander" for suggestion! $\endgroup$
    – Groups
    Jan 4, 2015 at 4:08
  • $\begingroup$ 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 $S_3$ ? $\endgroup$ Jan 4, 2015 at 10:11
  • 1
    $\begingroup$ 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? $\endgroup$
    – ahulpke
    Jan 5, 2015 at 16:57
  • $\begingroup$ OK! Does there exists subgroups of order $24$? What are they? $\endgroup$
    – Groups
    Jan 6, 2015 at 8:33

1 Answer 1


The comments specify that the question is specifically for subgroups of order 24. An explicit calculation shows that there are three such subgroups:

  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

they are groups number 6,7 and 8, respectively.


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