Subgroup ($H$) information
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,2)(3,6)(4,5)(7,8)(9,11)(10,12), (1,4)(2,5)(3,8)(6,7), (1,5,4,2)(3,6,8,7)(9,12,11,10), (9,11)(10,12)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $D_4^2:C_2$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{10}.D_6$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_4:C_4$ | ||||
| Normalizer: | $D_4^2:C_2$ | ||||
| Minimal over-subgroups: | $C_2^2\wr C_2$ | $C_2^2.D_4$ | $C_2^2\wr C_2$ | $C_4^2:C_2$ | $C_4\times D_4$ |
| Maximal under-subgroups: | $C_2^3$ | $C_2\times C_4$ | $C_2\times C_4$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $-8$ |
| Projective image | not computed |