Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(208\)\(\medspace = 2^{4} \cdot 13 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{39}, c^{26}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{78}:C_4^2$ |
| 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), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{52}:C_4$ |
| Order: | \(208\)\(\medspace = 2^{4} \cdot 13 \) |
| Exponent: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Automorphism Group: | $D_4\times F_{13}$, of order \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| Outer Automorphisms: | $C_3\times D_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{26}.(C_2^3\times C_6).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4992\)\(\medspace = 2^{7} \cdot 3 \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{78}:C_4^2$ | ||
| Normalizer: | $C_{78}:C_4^2$ | ||
| Complements: | $C_{52}:C_4$ | ||
| Minimal over-subgroups: | $C_{78}$ | $C_2\times C_6$ | $C_2\times C_6$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_{52}:C_4$ |