Subgroup ($H$) information
| Description: | $C_2^3:C_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, d^{19}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times C_4).D_{76}$ |
| Order: | \(1216\)\(\medspace = 2^{6} \cdot 19 \) |
| Exponent: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{19}$ |
| Order: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Exponent: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Automorphism Group: | $F_{19}$, of order \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
| Outer Automorphisms: | $C_9$, of order \(9\)\(\medspace = 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}.(C_{18}\times D_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| $W$ | $C_2^3:C_4$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_{38}$ | |
| Normalizer: | $(C_2\times C_4).D_{76}$ | |
| Minimal over-subgroups: | $C_2^3:C_{76}$ | $C_2^3.D_4$ |
| Maximal under-subgroups: | $C_2\times D_4$ | $C_2^2:C_4$ |
Other information
| Möbius function | $19$ |
| Projective image | $C_2^2.D_{76}$ |