Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $\He_7:(C_3^2\times D_6)$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $\He_7:(C_3^2\times D_6)$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $\He_7.(C_6\times S_3^2).C_2$ |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.(C_6\times S_3^2).C_2$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\He_7:(C_3^2\times D_6)$ | |||||||||
| Normalizer: | $\He_7:(C_3^2\times D_6)$ | |||||||||
| Complements: | $\He_7:(C_3^2\times D_6)$ | |||||||||
| Minimal over-subgroups: | $C_7$ | $C_7$ | $C_7$ | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_2$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\He_7:(C_3^2\times D_6)$ |