Subgroup ($H$) information
| Description: | $C_{11}\times D_{38}$ |
| Order: | \(836\)\(\medspace = 2^{2} \cdot 11 \cdot 19 \) |
| Index: | \(2\) |
| Exponent: | \(418\)\(\medspace = 2 \cdot 11 \cdot 19 \) |
| Generators: |
$a, bc^{11}, c^{22}, c^{266}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{22}\times D_{38}$ |
| Order: | \(1672\)\(\medspace = 2^{3} \cdot 11 \cdot 19 \) |
| Exponent: | \(418\)\(\medspace = 2 \cdot 11 \cdot 19 \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{19}\times A_4).C_{90}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{19}:(C_2^2\times C_{90})$ |
| $\card{\operatorname{res}(S)}$ | \(6840\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{19}$, of order \(38\)\(\medspace = 2 \cdot 19 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{38}$ |