Subgroup ($H$) information
| Description: | $C_5:D_5^2$ |
| Order: | \(500\)\(\medspace = 2^{2} \cdot 5^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(2,7,9,10,11)(3,8,12,15,4), (1,14,6,5,13)(3,8,12,15,4), (3,15)(7,11)(8,12)(9,10), (3,15,8,4,12), (5,6)(7,11)(9,10)(13,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_5^2\times C_{15}):S_4$ |
| Order: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, 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 \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $F_5\wr S_3$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $W$ | $C_5^3:S_4$, of order \(3000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Möbius function | $-27$ |
| Projective image | $(C_5^2\times C_{15}):S_4$ |