Subgroup ($H$) information
| Description: | $C_2^4:D_6^2$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,7)(6,8), (1,8,7)(2,4,3), (2,5)(3,4), (1,2)(3,6,4,7,5,8)(9,11)(10,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6:\SOPlus(4,2)$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $A_4^2.C_2^4.C_6.C_2^3$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $A_4^2:D_4^2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $S_4^2:C_2^2$ |