Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(7,8,9)(10,12,11), (4,6,5), (4,5,6)(10,11,12), (1,2,3)(4,6,5)(7,8,9)(10,12,11)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, abelian (hence metabelian and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^3:D_{12}$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^4:C_2$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(81\)\(\medspace = 3^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_3^3:D_{12}$ |