Subgroup ($H$) information
| Description: | $C_3\times D_{74}$ |
| Order: | \(444\)\(\medspace = 2^{2} \cdot 3 \cdot 37 \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(222\)\(\medspace = 2 \cdot 3 \cdot 37 \) |
| Generators: |
$a^{18}, b^{222}, b^{12}, b^{148}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{12}\times F_{37}$ |
| Order: | \(15984\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 37 \) |
| Exponent: | \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{18}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{222}.C_{18}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{37}:(C_2^2\times C_{36})$ |
| $W$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |||
| Normalizer: | $C_{12}\times F_{37}$ | |||
| Minimal over-subgroups: | $C_{222}:C_6$ | $C_{12}\times D_{37}$ | $C_{222}:C_4$ | |
| Maximal under-subgroups: | $C_{222}$ | $C_3\times D_{37}$ | $D_{74}$ | $C_2\times C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times F_{37}$ |