Subgroup ($H$) information
| Description: | $C_{13}^2:C_6$ |
| Order: | \(1014\)\(\medspace = 2 \cdot 3 \cdot 13^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$b^{6}c^{4}d^{11}, cd^{2}, d, b^{4}c^{7}d^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $F_{13}\wr C_2$, of order \(48672\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_{13}^2:(C_4\times S_3)$, of order \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_{13}^2:(C_4\times S_3)$ | ||
| Minimal over-subgroups: | $C_{13}^2:D_6$ | $C_{13}^2:C_{12}$ | $C_{13}^2:(C_3:C_4)$ |
| Maximal under-subgroups: | $C_{13}^2:C_3$ | $C_{13}:D_{13}$ | $C_{13}:C_6$ |
Other information
| Möbius function | $2$ |
| Projective image | $C_{13}^2:(C_4\times S_3)$ |