Subgroup ($H$) information
| Description: | $C_{13}:C_{12}$ |
| Order: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Index: | \(2\) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Generators: |
$a^{3}, a^{6}, b^{2}, a^{4}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_{13}:C_6$, of order \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{26}:C_6$ |