Subgroup ($H$) information
| Description: | $C_2^7$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,7)(3,4), (11,12)(14,16), (10,15)(11,12), (1,6)(5,8), (1,8)(5,6), (2,7)(3,4)(9,13)(10,15)(11,12)(14,16), (2,3)(4,7)\rangle$
|
| 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: | $A_4^2:C_2^4:C_4$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times C_3^2:C_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $\GL(7,2)$, of order \(163849992929280\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \cdot 127 \) |
| $W$ | $C_2\times C_3^2:C_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^3:(A_4^2:C_4)$ |