Subgroup ($H$) information
| Description: | $C_5^3:(C_{12}\times S_4)$ |
| Order: | \(36000\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$e^{2}, d^{6}f^{2}, c^{3}, c^{6}d^{24}e^{5}f^{4}, c^{6}d^{18}e^{8}f^{4}, d^{15}e^{9}f^{3}, ac^{6}d^{28}e^{8}f^{4}, d^{20}, b^{2}c^{8}de^{7}f^{2}, f$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_5^3:\He_3.C_2^3$ |
| Order: | \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $D_5^3:(C_4\times S_3)$, of order \(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \) |
| $W$ | $D_5^3.D_6$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $D_5^3:\He_3.C_2^3$ |