Subgroup ($H$) information
| Description: | $C_3^7.S_3\wr C_2^2$ |
| Order: | \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| Index: | $1$ |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$g^{3}, a^{3}f^{6}g^{3}, f^{9}, b^{3}d^{3}e^{11}f^{16}g^{6}, f^{8}g^{4}, g^{4}, a^{2}, f^{6}g^{3}, c^{3}d^{16}e^{14}f^{2}g^{2}, d^{6}e^{6}f^{12}, e^{6}, e^{14}g^{3}, e^{9}f^{11}g^{7}, d^{9}e^{11}f^{4}g^{6}, c^{2}d^{8}e^{10}f^{16}g^{3}, d^{8}e^{10}f^{10}g^{6}, b^{2}d^{5}e^{5}f^{6}g^{3}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a semidirect factor, nonabelian, and a Hall subgroup. Whether it is a direct factor or monomial 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.(C_3\times S_3\wr D_4)$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $W$ | $C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | not computed |
| 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 | $C_3^7.S_3\wr C_2^2$ |