Subgroup ($H$) information
| Description: | $C_3^2\times C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{6}d^{6}, b^{2}d^{6}, c^{4}, d^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and elementary for $p = 3$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $(C_2\times C_4).S_3^3$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $D_4:C_2^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Outer Automorphisms: | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^4.\SL(3,3)$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $\card{W}$ | \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |