Subgroup ($H$) information
| Description: | $C_2^3.D_6$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ad^{35}, d^{40}, d^{30}, b, c, c^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_5\times C_4^2.D_6$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2\times C_4\times C_2^6.C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_2^6:D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |