Properties

Label 124416.ch.8._.K
Order $ 2^{6} \cdot 3^{5} $
Index $ 2^{3} $
Normal No

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Subgroup ($H$) information

Description:$C_6^4:D_6$
Order: \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Generators: $\langle(1,2,9)(3,4,5)(6,7,8)(10,12,11)(13,17,24)(14,22,16)(15,23,21)(18,19,20) \!\cdots\! \rangle$ Copy content Toggle raw display
Derived length: $3$

The subgroup is nonabelian and monomial (hence solvable).

Ambient group ($G$) information

Description: $C_3^4.C_2\wr S_4$
Order: \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Exponent: \(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
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)$$C_3^3.C_4^2:A_4.D_6.C_2^5$
$\operatorname{Aut}(H)$ $C_6^4.C_6^2.C_2^2$
$\card{W}$ 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$4$
Möbius function not computed
Projective image not computed