Subgroup ($H$) information
| Description: | $C_6:D_{12}$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,5)(7,8)(11,13)(12,14), (1,5,2,3)(4,7,6,8), (1,4)(2,6)(3,8)(5,7), (9,11,13)(10,14,12), (1,2)(3,5)(4,6)(7,8), (10,14,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^3.D_6^2$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and rational.
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 \(2\) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.(C_2^4\times A_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times \SL(2,3).C_2^6$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \) |
| $\card{W}$ | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | not computed |