Subgroup ($H$) information
| Description: | $C_{12}\times D_{38}$ |
| Order: | \(912\)\(\medspace = 2^{4} \cdot 3 \cdot 19 \) |
| Index: | \(2\) |
| Exponent: | \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \) |
| Generators: |
$c^{76}, b^{2}, c^{12}, c^{114}, b, a$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_4\times C_{12}\times D_{19}$ |
| Order: | \(1824\)\(\medspace = 2^{5} \cdot 3 \cdot 19 \) |
| Exponent: | \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_{114}.C_9.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{38}.C_{18}.C_2^5$ |
| $\card{\operatorname{res}(S)}$ | \(21888\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 19 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{19}$, of order \(38\)\(\medspace = 2 \cdot 19 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $D_{38}$ |