Subgroup ($H$) information
| Description: | $C_2\times C_{74}$ |
| Order: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Generators: |
$ab^{4}, c^{37}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_2\times C_{74}:C_8$ |
| Order: | \(1184\)\(\medspace = 2^{5} \cdot 37 \) |
| Exponent: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $Q_8:C_2^2.C_{37}.(C_{36}\times S_3)$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_{36}$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $S_3\times C_{36}$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{37}:C_8$ |