Subgroup ($H$) information
| Description: | $C_2^2\times D_{10}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(1,4)(2,3)(5,6)(7,8)(9,12)(10,13), (1,2)(3,4)(9,12,13,11,10), (1,3)(2,4)(5,7)(6,8), (1,4)(2,3), (1,2)(3,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_4\times A_4\times A_5$ |
| Order: | \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times S_4\times S_5$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $F_5\times C_2^3:\GL(3,2)$, of order \(26880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7 \) |
| $W$ | $C_3\times D_5$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $D_4\times A_4\times A_5$ |