Subgroup ($H$) information
| Description: | $C_5^2:C_{12}$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$a^{2}, a^{4}, cd^{3}, b, d^{3}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_5\times C_{60}).S_3$ |
| Order: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_5^2:(C_2\times C_4\times S_3)$ |
| $\operatorname{Aut}(H)$ | $F_{25}:C_2^2$, of order \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_5^2:(C_4\times D_6)$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_5^2:S_3$, of order \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $(C_5\times C_{60}).S_3$ | ||
| Minimal over-subgroups: | $C_3\times C_5^2:C_{12}$ | $(C_5\times C_{20}).S_3$ | |
| Maximal under-subgroups: | $C_5^2:C_6$ | $C_5\times C_{20}$ | $C_{12}$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_3\times C_5^2:S_3$ |