Subgroup ($H$) information
| Description: | $C_4^2:C_2^3$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(2,5)(3,4)(10,11)(12,13)(14,15), (2,5)(3,4), (12,13)(14,15), (2,3,5,4)(6,11) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times \GL(2,\mathbb{Z}/4):F_5$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^4.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^8.C_2^6.S_4.C_2$ |
| $\operatorname{res}(S)$ | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $15$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times F_5\times S_4$ |