Subgroup ($H$) information
| Description: | $C_{44}:Q_8$ |
| Order: | \(352\)\(\medspace = 2^{5} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$a, c^{22}, c^{33}, c^{4}, b^{2}c^{11}, b^{4}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, nonabelian, elementary for $p = 2$ (hence hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{44}.D_8$ |
| Order: | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.(C_2^5\times C_{10}).C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{10}\times C_2^4.C_2^4.C_2$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}\times D_4^2$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2\times C_{22}$ | |||||
| Normalizer: | $C_{44}.D_8$ | |||||
| Minimal over-subgroups: | $C_{44}.D_8$ | |||||
| Maximal under-subgroups: | $C_4\times C_{44}$ | $C_4:C_{44}$ | $C_4:C_{44}$ | $C_4:C_{44}$ | $Q_8\times C_{22}$ | $C_4:Q_8$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{22}.D_4$ |