Subgroup ($H$) information
| Description: | $C_{136}$ |
| Order: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Generators: |
$c, d^{2}, c^{4}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3:A_4.C_8.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(S)$ | $C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $\OD_{16}:C_{34}$ | ||
| Normalizer: | $C_{136}.C_2^3$ | ||
| Minimal over-subgroups: | $C_2\times C_{136}$ | $\OD_{16}\times C_{17}$ | $\OD_{16}\times C_{17}$ |
| Maximal under-subgroups: | $C_{68}$ | $C_8$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $-8$ |
| Projective image | $C_2^4$ |