Subgroup ($H$) information
| Description: | $C_5^3:C_2$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$c^{6}d^{18}e^{8}f^{4}, e^{2}f^{3}, f, d^{6}f$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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.
Quotient group ($Q$) structure
| Description: | $C_2\times C_6^2:D_6$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $\AGL(3,5)$, of order \(186000000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{6} \cdot 31 \) |
| $W$ | $D_5^3.D_6$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
Related subgroups
| Centralizer: | $C_3:S_3$ | |
| Normalizer: | $D_5^3:\He_3.C_2^3$ | |
| Minimal over-subgroups: | $C_5^3:C_6$ | $C_5^3:C_4$ |
| Maximal under-subgroups: | $C_5^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $D_5^3:\He_3.C_2^3$ |