Subgroup ($H$) information
| Description: | $C_2^3.D_6^2$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(5,10)(11,14), (2,6,3), (5,8)(9,14)(10,12)(11,13), (5,11)(8,9)(10,14)(12,13) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^4:D_6^2$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| 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: | $C_2$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_4:S_6$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_6^2.(C_2^4\times A_4).C_2^3$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times S_7$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_6^2:D_4$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $D_6^2:D_4$ |