Subgroup ($H$) information
| Description: | $C_4\times S_3$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ac^{5}d^{9}e^{3}f, b^{3}c^{9}d^{21}e^{2}f^{4}, c^{6}d^{18}e^{4}f^{2}, b^{2}d^{18}e^{4}f^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $D_5^3:\He_3.C_2^3$ |
| Order: | \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | |
| Normalizer: | $C_4\times D_6$ | |
| Normal closure: | $D_5^3.C_3^2:D_6$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_4\times S_4$ | $C_4\times D_6$ |
| Maximal under-subgroups: | $C_2\times C_4$ |
Other information
| Number of subgroups in this autjugacy class | $4500$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $D_5^3:\He_3.C_2^3$ |