Subgroup ($H$) information
| Description: | $D_5\times \He_3$ |
| Order: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{3}, c^{10}, b^{4}, a^{2}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{30}.C_6^2$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \AGL(2,3)\times F_5$ |
| $\operatorname{Aut}(H)$ | $F_5\times C_3^2:\GL(2,3)$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5\times C_3^2:\GL(2,3)$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^2\times D_5$, of order \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $C_{30}.C_6^2$ | ||
| Complements: | $C_4$ $C_4$ | ||
| Minimal over-subgroups: | $D_{10}\times \He_3$ | ||
| Maximal under-subgroups: | $C_5\times \He_3$ | $C_3^2\times D_5$ | $C_2\times \He_3$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_{60}:C_6$ |