Subgroup ($H$) information
| Description: | $C_2\times C_{12}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(13\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{3}, b^{13}, a^{4}, a^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{26}:C_{12}$ |
| Order: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times F_{13}$, of order \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $13$ |
| Möbius function | $-1$ |
| Projective image | $C_{13}:C_6$ |