Subgroup ($H$) information
| Description: | $C_2^7$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(2\) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rr}
1 & 12 \\
18 & 19
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
6 & 23
\end{array}\right), \left(\begin{array}{rr}
17 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
19 & 12 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
7 & 0 \\
18 & 17
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
7 & 12 \\
0 & 7
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^5:D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{15}.C_2^4.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $\GL(7,2)$, of order \(163849992929280\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \cdot 127 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^9.C_2^6:\GL(3,2)$, of order \(5505024\)\(\medspace = 2^{18} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2^7$ | ||
| Normalizer: | $C_2^5:D_4$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $C_2^5:D_4$ | ||
| Maximal under-subgroups: | $C_2^6$ | $C_2^6$ | $C_2^6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^4$ |