Subgroup ($H$) information
| Description: | not computed |
| Order: | \(39366\)\(\medspace = 2 \cdot 3^{9} \) |
| Index: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$c^{3}d^{16}e^{14}f^{2}g^{2}, f^{6}, f^{2}g^{3}, g, g^{3}, a^{2}c^{4}d^{4}e^{14}f^{2}g^{3}, e^{2}f^{2}g^{2}, e^{6}f^{6}g^{6}, d^{6}e^{12}f^{6}g^{6}, d^{8}e^{10}f^{2}g^{2}$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), 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: | $D_6^2:C_2$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^2.C_2^6.C_2^3$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2^3:D_4$, of order \(64\)\(\medspace = 2^{6} \) |
| 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 |
| $W$ | $C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | $C_1$ |
| 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 | $C_3^7.S_3\wr C_2^2$ |