Subgroup ($H$) information
| Description: | $D_{26}:C_6$ |
| Order: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$a^{2}, c^{13}, b^{4}, b^{6}, c^{2}$
|
| 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: | $C_2\times C_4\times F_{13}$ |
| Order: | \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| 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: | $C_2^2$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\wr C_2^2\times F_{13}$, of order \(9984\)\(\medspace = 2^{8} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $S_4\times F_{13}$, of order \(3744\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 13 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times F_{13}$, of order \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_2\times F_{13}$ |