Subgroup ($H$) information
| Description: | $C_{19}^2:C_6$ |
| Order: | \(2166\)\(\medspace = 2 \cdot 3 \cdot 19^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(114\)\(\medspace = 2 \cdot 3 \cdot 19 \) |
| Generators: |
$b^{2}c^{11}d^{13}, a^{6}, d, cd^{17}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), 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: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}^2.C_{36}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{19}^2.\GL(2,19)$, of order \(44446320\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \cdot 19^{3} \) |
| $W$ | $D_{19}^2:C_{18}$, of order \(25992\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2} \) |
Related subgroups
Other information
| Möbius function | $-2$ |
| Projective image | $D_{19}^2:C_{18}$ |