Subgroup ($H$) information
| Description: | $Q_8:A_4:S_3^2$ |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,5)(2,6)(7,8,9,14,13,12)(10,11), (7,12)(8,10)(9,14)(11,13), (2,4,6), (9,12) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $D_4:C_6^2:D_{12}$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.A_4^2.C_4.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_3^5:D_6$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| $W$ | $(C_2^2\times C_6^2):D_{12}$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $(C_2^2\times C_6^2):D_{12}$ |