Subgroup ($H$) information
| Description: | $C_{13}:C_4$ |
| Order: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Generators: |
$a^{3}b^{3}c, c^{2}, a^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{26}:C_{12}$ |
| Order: | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\wr C_2\times F_{13}$, of order \(4992\)\(\medspace = 2^{7} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_{13}:C_6$, of order \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
Related subgroups
Other information
| Möbius function | $-2$ |
| Projective image | $D_{26}:C_6$ |