Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,2)(3,4)(5,6)(7,8)(9,10)(11,13), (1,3)(2,4), (1,4)(2,3)(5,8)(6,7), (1,2)(3,4)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $C_2^4\times A_5$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times A_8$, of order \(2419200\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^6$ | |||
| Normalizer: | $C_2^6$ | |||
| Normal closure: | $C_2^4\times A_5$ | |||
| Core: | $C_2^3$ | |||
| Minimal over-subgroups: | $C_2^2\times D_{10}$ | $C_2^2\times D_6$ | $C_2^5$ | $C_2^5$ |
| Maximal under-subgroups: | $C_2^3$ | $C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $225$ |
| Number of conjugacy classes in this autjugacy class | $15$ |
| Möbius function | $0$ |
| Projective image | $C_2\times A_5$ |