Subgroup ($H$) information
| Description: | $C_{37}$ |
| Order: | \(37\) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(37\) |
| Generators: |
$c^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $37$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_4.D_{148}$ |
| 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 $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4.D_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{74}.C_{18}.C_4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2368\)\(\medspace = 2^{6} \cdot 37 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $\OD_{16}\times C_{37}$ | |
| Normalizer: | $C_4.D_{148}$ | |
| Minimal over-subgroups: | $C_{74}$ | $C_{74}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_4.D_{148}$ |