Subgroup ($H$) information
| Description: | $C_2^5:D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(3\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(9,11)(10,12), (2,6)(7,8), (1,3)(2,8)(4,5)(6,7)(10,12), (1,4)(2,6)(3,5) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, nonabelian, a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^7:C_6$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:C_3.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^{15}.C_2^4.\PSL(2,7)$ |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | not computed |