Subgroup ($H$) information
| Description: | $C_5^6:(C_2\times F_7)$ |
| Order: | \(1312500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{6} \cdot 7 \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(2,5)(3,4)(6,8)(9,10)(11,13)(14,15)(16,18)(19,20)(21,23)(24,25)(26,29)(27,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^7:(C_2\times \PGL(2,7))$ |
| Order: | \(52500000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 40T188410.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^7:(C_4\times \PGL(2,7))$, of order \(105000000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{7} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_5^6.C_8:F_7$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $40$ |
| Möbius function | not computed |
| Projective image | not computed |