Subgroup ($H$) information
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{4}, b^{37}$
|
| 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 metacyclic.
Ambient group ($G$) information
| Description: | $C_{148}:C_8$ |
| Order: | \(1184\)\(\medspace = 2^{5} \cdot 37 \) |
| Exponent: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{37}:C_4$ |
| Order: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
| Exponent: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
| Automorphism Group: | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| Outer Automorphisms: | $C_9$, of order \(9\)\(\medspace = 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{74}.C_{18}.C_4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10656\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 37 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{148}:C_4$ | ||
| Normalizer: | $C_{148}:C_8$ | ||
| Minimal over-subgroups: | $C_2\times C_{148}$ | $C_4^2$ | |
| Maximal under-subgroups: | $C_2^2$ | $C_4$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{74}:C_4$ |