Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(2\) |
| Generators: |
$e, b^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2^3\times C_8^2$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_4\times C_8$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^8.C_2\wr D_6$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| Outer Automorphisms: | $C_2^8.C_2\wr D_6$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_2.C_4^2).C_2^6.C_2^6.C_3.D_4.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12582912\)\(\medspace = 2^{22} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_8^2$ | |||
| Normalizer: | $C_2^3\times C_8^2$ | |||
| Minimal over-subgroups: | $C_2\times C_4$ | $C_2\times C_4$ | $C_2^3$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $42$ |
| Number of conjugacy classes in this autjugacy class | $42$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times C_4\times C_8$ |