Subgroup ($H$) information
| Description: | $C_{860}$ |
| Order: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Generators: |
$b^{11610}, b^{18576}, b^{23220}, b^{1080}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,43$), 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: | $D_{54}$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Automorphism Group: | $C_{54}:C_{18}$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| 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 \(2612736\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{84}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\card{W}$ | 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 |