Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{4}, c^{34}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $Q_8\times C_{136}$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_{68}$ |
| Order: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Exponent: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Automorphism Group: | $C_2^3.(C_8\times S_4).C_2$ |
| Outer Automorphisms: | $C_2^3.(C_8\times S_4).C_2$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{16}\times C_2^4:C_3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_2^2\times C_{68}$ |