Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
9 & 12 \\
0 & 9
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_4^2.C_2^5$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| 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: | $C_2^5\times C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| Outer Automorphisms: | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^{14}\times C_4).C_2^6.C_2^2.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_2^3\times C_4\times C_8$ | |||
| Normalizer: | $C_4^2.C_2^5$ | |||
| Minimal over-subgroups: | $C_2\times C_4$ | $C_2\times C_4$ | $D_4$ | $C_8$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |