Subgroup ($H$) information
| Description: | $C_8:D_{22}$ |
| Order: | \(352\)\(\medspace = 2^{5} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Generators: |
$ab, d, c, c^{6}, b^{2}c^{2}, c^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times C_8).D_{22}$ |
| 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 $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times C_{11}:C_5).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_{22}.(C_2^4\times C_{10}).C_2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2\times D_{22}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_4\times D_{11}$ |