Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(15708\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$b^{23562}, b^{15708}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11781}:Q_8$ |
| Order: | \(94248\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(47124\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{102}\times D_{77}$ |
| Order: | \(15708\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(7854\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 17 \) |
| Automorphism Group: | $C_{77}.C_{240}.C_2^4$ |
| Outer Automorphisms: | $C_2^3\times C_{240}$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(3548160\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |