Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group (for every $p$), perfect, and rational. Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3\wr C_2^2$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_3^6.C_3\wr C_2^2$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| Outer Automorphisms: | $D_6\times C_3^3$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^6.C_3\wr C_2^2$ |
| Normalizer: | $C_3^6.C_3\wr C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_3\wr C_2^2$ |