Subgroup ($H$) information
| Description: | $C_{68}$ |
| Order: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Generators: |
$c^{2}, d^{2}, c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and cyclic (hence elementary ($p = 2,17$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_{136}.C_2^3$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| 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), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3:A_4.C_8.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{136}.C_2^3$ | ||
| Normalizer: | $C_{136}.C_2^3$ | ||
| Minimal over-subgroups: | $C_2\times C_{68}$ | $C_2\times C_{68}$ | $C_{136}$ |
| Maximal under-subgroups: | $C_{34}$ | $C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $64$ |
| Projective image | $C_2^4$ |