Subgroup ($H$) information
| Description: | $C_3^6$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$a^{2}, g, e, b^{6}, c^{2}, df^{2}g$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5.S_3^3$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| 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: | $S_3\times D_6$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6\wr C_2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_4^2:C_2^2)$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $\GL(6,3)$, of order \(84129611558952960\)\(\medspace = 2^{13} \cdot 3^{15} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \) |
| $W$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^7$ |
| Normalizer: | $C_3^5.S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5.S_3^3$ |