Subgroup ($H$) information
| Description: | $C_{11610}$ |
| Order: | \(11610\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \cdot 43 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(11610\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \cdot 43 \) |
| Generators: |
$b^{1080}, b^{10320}, b^{30960}, b^{18576}, b^{3440}, b^{23220}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{215}\times D_{216}$ |
| Order: | \(92880\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 43 \) |
| Exponent: | \(46440\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \cdot 43 \) |
| 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: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(2612736\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_6\times C_{252}$ |
| $\card{W}$ | 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 |