Subgroup ($H$) information
| Description: | $C_{13}:C_6$ |
| Order: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$c^{13}, c^{2}, b^{4}$
|
| 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 $p = 3$.
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: | $C_4^2$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\wr C_2^2\times F_{13}$, of order \(9984\)\(\medspace = 2^{8} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | ||
| Normalizer: | $C_2\times C_4\times F_{13}$ | ||
| Complements: | $C_4^2$ | ||
| Minimal over-subgroups: | $C_{26}:C_6$ | $C_{26}:C_6$ | |
| Maximal under-subgroups: | $C_{13}:C_3$ | $C_{26}$ | $C_6$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_4\times F_{13}$ |