Subgroup ($H$) information
| Description: | $C_{13}^2:S_3$ |
| Order: | \(1014\)\(\medspace = 2 \cdot 3 \cdot 13^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$ab^{7}c^{2}d^{6}, cd^{6}, d, b^{4}c^{4}d^{12}$
|
| Derived length: | $3$ |
The subgroup is normal, a semidirect factor, nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{13}^2:(C_4\times S_3)$ |
| Order: | \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{13}^2:(S_3\times C_{12})$, of order \(12168\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{res}(S)$ | $C_{13}^2:(S_3\times C_{12})$, of order \(12168\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_{13}^2:(C_4\times S_3)$, of order \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{13}^2:(C_4\times S_3)$ |