Subgroup ($H$) information
| Description: | $C_{19}^2:C_4$ |
| Order: | \(1444\)\(\medspace = 2^{2} \cdot 19^{2} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Generators: |
$b, b^{2}, d, cd^{15}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_{19}^2:C_{18}$ |
| Order: | \(25992\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2} \) |
| Exponent: | \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_{18}$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}^2.C_{36}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{19}^2.C_{360}.C_2$ |
| $W$ | $D_{19}^2:C_{18}$, of order \(25992\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $D_{19}^2:C_{18}$ | |
| Complements: | $C_{18}$ $C_{18}$ | |
| Minimal over-subgroups: | $C_{19}^2:C_{12}$ | $D_{19}\wr C_2$ |
| Maximal under-subgroups: | $C_{19}:D_{19}$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{19}^2:C_{18}$ |