Subgroup ($H$) information
| Description: | $(C_5^2\times C_{15}):S_4$ |
| Order: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Index: | \(2\) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,8)(3,5,4)(6,7,12)(9,11,10)(13,14,15)(16,17,18), (2,14,5,11,7)(4,12,15,10,8) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_5^3:(S_3\times S_4)$ |
| Order: | \(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times D_5^3.D_6$, of order \(72000\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $W$ | $C_5^3:(S_3\times S_4)$, of order \(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_5^3:(S_3\times S_4)$ |