Subgroup ($H$) information
Description: | $C_5\times A_6$ |
Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Index: | \(4\)\(\medspace = 2^{2} \) |
Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Generators: |
$\langle(1,3,4,2,5)(6,15)(7,12)(8,9)(11,13), (1,5,2,4,3)(6,10,9,7)(12,15,13,14), (1,2,3,5,4)\rangle$
|
Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), nonabelian, and nonsolvable.
Ambient group ($G$) information
Description: | $S_6:C_{10}$ |
Order: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
Description: | $C_2^2$ |
Order: | \(4\)\(\medspace = 2^{2} \) |
Exponent: | \(2\) |
Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
$\operatorname{Aut}(H)$ | $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
$W$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Centralizer: | $C_5$ | |||
Normalizer: | $S_6:C_{10}$ | |||
Minimal over-subgroups: | $C_5\times S_6$ | $C_5\times \PGL(2,9)$ | $A_6.C_{10}$ | |
Maximal under-subgroups: | $A_6$ | $C_5\times A_5$ | $C_3^2:C_{20}$ | $C_5\times S_4$ |
Other information
Möbius function | $2$ |
Projective image | $S_6:C_2$ |