Subgroup ($H$) information
| Description: | $C_5:D_5$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$ab, c^{3}, b^{2}$
|
| 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: | $C_3\times D_5^2$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5^2:C_2^2$, of order \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_5^2.\GL(2,5)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_5\wr C_2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_5^2$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $C_3\times D_5^2$ | ||||
| Complements: | $C_6$ $C_6$ | ||||
| Minimal over-subgroups: | $C_{15}:D_5$ | $D_5^2$ | |||
| Maximal under-subgroups: | $C_5^2$ | $D_5$ | $D_5$ | $D_5$ | $D_5$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_3\times D_5^2$ |