Subgroup ($H$) information
| Description: | $C_3^4:S_3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | $1$ |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{206} & \alpha^{147} \\ \alpha^{126} & \alpha^{92} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{115} & \alpha^{203} \\ \alpha^{182} & \alpha^{95} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{189} & \alpha^{196} \\ \alpha^{175} & \alpha^{241} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{203} & \alpha^{134} \\ \alpha^{113} & \alpha^{40} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{121} & 0 \\ \alpha^{171} & 1 \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{88} & \alpha^{141} \\ \alpha^{120} & \alpha^{10} \\ \end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_3^4:S_3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\AGL(5,3)$, of order \(115562653240320\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $\AGL(5,3)$, of order \(115562653240320\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $W$ | $C_3^4:S_3$, of order \(486\)\(\medspace = 2 \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $C_3^4:S_3$ | |
| Complements: | $C_1$ | |
| Maximal under-subgroups: | $C_3^5$ | $C_3^3:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^4:S_3$ |