Properties

Label 1814400.a.8400.a1.a1
Order $ 2^{3} \cdot 3^{3} $
Index $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Normal No

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Subgroup ($H$) information

Description:$C_3^2:S_4$
Order: \(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Index: \(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Generators: $\langle(2,3,4)(5,9,7), (2,4,3), (1,3)(2,4), (1,2)(3,4), (1,2,3,4)(5,8)(6,7)(9,10), (1,3,2)(6,10,8)\rangle$ Copy content Toggle raw display
Derived length: $3$

The subgroup is nonabelian and monomial (hence solvable).

Ambient group ($G$) information

Description: $A_{10}$
Order: \(1814400\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7 \)
Exponent: \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Derived length:$0$

The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

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)$$S_{10}$, of order \(3628800\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7 \)
$\operatorname{Aut}(H)$ $C_2\times C_6^2:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
$W$$S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)

Related subgroups

Centralizer:$C_3$
Normalizer:$A_4:S_3^2$
Normal closure:$A_{10}$
Core:$C_1$
Minimal over-subgroups:$A_4:S_3^2$
Maximal under-subgroups:$C_3^2\times A_4$$C_6\wr C_2$$C_3\times S_4$$C_3\times S_4$$C_3:S_4$$C_3^2:C_6$

Other information

Number of subgroups in this conjugacy class$4200$
Möbius function$0$
Projective image$A_{10}$