Properties

Label 6718464.bb.6561.a1
Order $ 2^{10} $
Index $ 3^{8} $
Normal No

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Subgroup ($H$) information

Description:$(C_2\times D_4^2):D_4$
Order: \(1024\)\(\medspace = 2^{10} \)
Index: \(6561\)\(\medspace = 3^{8} \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Generators: $\langle(14,15)(16,17), (1,9)(2,7)(3,8)(4,11)(5,12)(6,10), (1,15,2,14)(3,13)(4,17,5,16) \!\cdots\! \rangle$ Copy content Toggle raw display
Nilpotency class: $4$
Derived length: $3$

The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and rational.

Ambient group ($G$) information

Description: $C_3^8:D_4^2.(C_2\times D_4)$
Order: \(6718464\)\(\medspace = 2^{10} \cdot 3^{8} \)
Exponent: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:$4$

The ambient group is nonabelian, solvable, and rational. 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_3^8.C_2^4.C_2^4.D_4^2$, of order \(107495424\)\(\medspace = 2^{14} \cdot 3^{8} \)
$\operatorname{Aut}(H)$ $C_2^9.C_2\wr D_4$, of order \(65536\)\(\medspace = 2^{16} \)
$\card{W}$ not computed

Related subgroups

Centralizer: not computed
Normalizer:$(C_2\times D_4^2):D_4$
Normal closure:$C_3^8:D_4^2.(C_2\times D_4)$
Core:$C_1$

Other information

Number of subgroups in this autjugacy class$6561$
Number of conjugacy classes in this autjugacy class$1$
Möbius function not computed
Projective image not computed