Subgroup ($H$) information
| Description: | $C_{172}$ |
| Order: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Generators: |
$c^{86}, c^{8}, c^{172}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_2^2\times C_{344}$ |
| Order: | \(1376\)\(\medspace = 2^{5} \cdot 43 \) |
| Exponent: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| 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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| 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), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3:A_4.C_{42}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $C_2^3$ |