Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Exponent: | \(2\) |
| Generators: |
$a^{2}, b^{2}c^{19}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_{38}.C_4^2$ |
| Order: | \(608\)\(\medspace = 2^{5} \cdot 19 \) |
| Exponent: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Nilpotency class: | $2$ |
| 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: | $D_4\times C_{19}$ |
| Order: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Exponent: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Automorphism Group: | $D_4\times C_{18}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2\times C_{18}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{18}\times C_2^4:C_3.D_4$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_4\times C_{19}$ |