Kourovka Notebook Problem 20.21

Definitions and mathematical statement

Let $C_{12}={\mathbb{Z}}/12{\mathbb{Z}}$ denote the cyclic group of order $12$, and let $A_4$ denote the group of even permutations of $\{1,2,3,4\}$. For a subgroup $H\leq G$, write $[G:H]$ for its index, and write $H\trianglelefteq G$ to mean that $H$ is normal in $G$.

Does there exist a finite group $G$ with normal subgroups $K,L\trianglelefteq G$ such that \[ [G:K]=[G:L]=12,\qquad K\cong L,\qquad G/K\cong C_{12},\qquad G/L\cong A_4? \] Here $K\cong L$ means isomorphism as abstract groups.

The supplied statement bears the attribution "G. Verret", whereas the background and proposer metadata name M. Conder. Their respective roles are unresolved in the supplied record; this attribution discrepancy does not change the stated existence question.

Short English statement

Can a finite group have two isomorphic normal subgroups, each of index $12$, with one quotient cyclic of order $12$ and the other isomorphic to $A_4$? The supplied registry labels this problem open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, supplied complete source record with permanent ID $701679$, source code catalogue entry, titled Kourovka Notebook Problem 20.21.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem number 20.21; the background identifies the 20th issue (2022) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The collection metadata names E. I. Khukhro and V. D. Mazurov as editors.
• The background and proposer metadata attribute the problem to M. Conder, with proposed year 2022; the statement itself begins with the attribution "G. Verret". The supplied record does not explain these respective roles.