Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(6,8)(7,11), (3,4)(6,8)(7,11)(10,12), (6,7)(8,11), (1,2)(3,12)(4,10)(5,9)(6,11)(7,8), (1,5)(2,9)(6,11)(7,8), (3,12)(4,10)(6,7)(8,11)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^2:A_4^2:S_3^2$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| 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: | $C_3^2:S_3^2$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Outer Automorphisms: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.\He_3.Q_8.S_3^2$ |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $W$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2:A_4^2:S_3^2$ |