Subgroup ($H$) information
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
352 & 0 \\
0 & 352
\end{array}\right), \left(\begin{array}{rr}
136 & 0 \\
0 & 136
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $D_8.C_{88}$ |
| Order: | \(1408\)\(\medspace = 2^{7} \cdot 11 \) |
| Exponent: | \(176\)\(\medspace = 2^{4} \cdot 11 \) |
| Nilpotency class: | $3$ |
| 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: | $C_8\times D_4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| Outer Automorphisms: | $D_4\times C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:((C_2\times C_4).C_2^6)$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $D_8.C_{88}$ | ||||||
| Normalizer: | $D_8.C_{88}$ | ||||||
| Minimal over-subgroups: | $C_2\times C_{22}$ | $C_{44}$ | $C_{44}$ | $C_2\times C_{22}$ | $C_2\times C_{22}$ | $C_{44}$ | $C_{44}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_8\times D_4$ |