Subgroup ($H$) information
| Description: | $C_4:Q_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ab, b^{2}c^{10}, c^{5}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_{20}.\SD_{16}$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| 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: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_4^2:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4^2:C_2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2\times C_{10}$ | ||||
| Normalizer: | $C_{20}.\SD_{16}$ | ||||
| Minimal over-subgroups: | $C_{20}:Q_8$ | $Q_8:Q_8$ | |||
| Maximal under-subgroups: | $C_4^2$ | $C_4:C_4$ | $C_4:C_4$ | $C_2\times Q_8$ | $C_4:C_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_4\times C_{10}$ |