Subgroup ($H$) information
| Description: | $C_4^2:C_2^2$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(2\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
13 & 10 \\
4 & 3
\end{array}\right), \left(\begin{array}{rr}
7 & 6 \\
1 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
8 & 9
\end{array}\right), \left(\begin{array}{rr}
5 & 8 \\
12 & 13
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_4^2:D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3.C_2^6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^5.D_4^2$, of order \(2048\)\(\medspace = 2^{11} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2 . C_2^7$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^2\wr C_2$ |