Subgroup ($H$) information
| Description: | $C_{318}$ |
| Order: | \(318\)\(\medspace = 2 \cdot 3 \cdot 53 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(318\)\(\medspace = 2 \cdot 3 \cdot 53 \) |
| Generators: |
$a, c^{212}, c^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,53$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_2^2\times C_{318}$ |
| Order: | \(1272\)\(\medspace = 2^{3} \cdot 3 \cdot 53 \) |
| Exponent: | \(318\)\(\medspace = 2 \cdot 3 \cdot 53 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{52}\times \PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{52}$, of order \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{52}$, of order \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_2^2$ |