Subgroup ($H$) information
| Description: | not computed |
| Order: | \(19683\)\(\medspace = 3^{9} \) |
| Index: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$a^{2}c^{4}d^{4}e^{14}f^{2}g^{3}, g^{4}, d^{2}e^{8}f^{8}g^{4}, e^{8}f^{8}g^{8}, f^{8}g^{3}$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.S_3\wr C_2^2$ |
| Order: | \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4:S_3^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $(C_2^3\times S_3^2).C_2^4$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(C_3\times S_3\wr D_4)$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^7.S_3\wr C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.S_3\wr C_2^2$ |