Subgroup ($H$) information
| Description: | $D_5^3:\He_3$ |
| Order: | \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$f^{3}, ef^{12}, d^{20}, d^{15}ef^{6}, f^{10}, d^{6}f^{6}, b^{6}, b^{4}d^{4}e^{4}f^{14}, cd^{27}e^{2}f^{6}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $D_5^3.C_3^2:D_6$ |
| Order: | \(108000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \) |
| 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: | $C_2^2$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_5^3.C_6^2.(C_4\times S_3^2)$ |
| $W$ | $C_5^3:(C_{12}\times S_4)$, of order \(36000\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $D_5^3.C_3^2:D_6$ |