Subgroup ($H$) information
| Description: | $C_{19}$ |
| Order: | \(19\) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(19\) |
| Generators: |
$c^{4}$
|
| 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 $19$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{38}.C_4^2$ |
| Order: | \(608\)\(\medspace = 2^{5} \cdot 19 \) |
| Exponent: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2.C_4^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $2$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{38}).C_{18}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2432\)\(\medspace = 2^{7} \cdot 19 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2^2\times C_{76}$ | ||||||
| Normalizer: | $C_{38}.C_4^2$ | ||||||
| Minimal over-subgroups: | $C_{38}$ | $C_{38}$ | $C_{38}$ | $C_{38}$ | $C_{38}$ | $C_{38}$ | $C_{38}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{38}.C_4^2$ |