Subgroup ($H$) information
| Description: | $C_{38}$ |
| Order: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Generators: |
$b^{4}, c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,19$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{76}.D_8$ |
| Order: | \(1216\)\(\medspace = 2^{6} \cdot 19 \) |
| Exponent: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Nilpotency class: | $3$ |
| 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: | $Q_8:C_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $D_4\times C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2^3$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{18}\times D_4^2$ |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{76}.D_8$ | ||
| Normalizer: | $C_{76}.D_8$ | ||
| Minimal over-subgroups: | $C_2\times C_{38}$ | $C_{76}$ | $C_{76}$ |
| Maximal under-subgroups: | $C_{19}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $Q_8:C_4$ |