Subgroup ($H$) information
| Description: | $C_{193}:C_{12}$ |
| Order: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Index: | \(2\) |
| Exponent: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Generators: |
$a^{3}b, a^{2}, b^{4}, b^{386}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{12}\times D_{193}$ |
| Order: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Exponent: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_2\times C_4)$, of order \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{193}.C_{96}.C_2^3$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_5:F_5^2$, of order \(4000\)\(\medspace = 2^{5} \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{193}$, of order \(386\)\(\medspace = 2 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $C_{12}\times D_{193}$ | ||
| Complements: | $C_2$ $C_2$ | ||
| Minimal over-subgroups: | $C_{12}\times D_{193}$ | ||
| Maximal under-subgroups: | $C_{1158}$ | $C_{193}:C_4$ | $C_{12}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_{386}$ |