Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(14,16)(15,17), (14,15)(16,17)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $D_4\times C_6^2.S_3^2$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6^3:D_6$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_6^3:(C_6\times D_6)$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_6^2).C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_6^2.D_6^2$ | ||||||
| Normalizer: | $D_4\times C_6^2.S_3^2$ | ||||||
| Complements: | $C_6^3:D_6$ | ||||||
| Minimal over-subgroups: | $C_2\times C_6$ | $D_4$ | $D_4$ | $D_4$ | $D_4$ | $D_4$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_6^2.D_6^2$ |