Subgroup ($H$) information
| Description: | $C_2^7$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$g, a^{2}, b^{2}ce, f, b^{2}ef, cd, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), 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: | $C_2^6.D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^2\wr C_2^2$, of order \(262144\)\(\medspace = 2^{18} \) |
| $\operatorname{Aut}(H)$ | $\GL(7,2)$, of order \(163849992929280\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \cdot 127 \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^7$ | ||||||||
| Normalizer: | $C_2^6.D_4$ | ||||||||
| Minimal over-subgroups: | $C_2^5:D_4$ | $C_2^6:C_4$ | |||||||
| Maximal under-subgroups: | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ | $C_2^6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | not computed |