Properties

Label 95040.a.7920.b1.a1
Order $ 2^{2} \cdot 3 $
Index $ 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Normal No

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Subgroup ($H$) information

Description:$C_2\times C_6$
Order: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Index: \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Generators: $\langle(1,2)(3,4)(5,9)(6,11)(7,8)(10,12), (1,9)(2,5)(3,8)(4,7)(6,10)(11,12), (1,11,8)(2,6,7)(3,9,12)(4,5,10)\rangle$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Ambient group ($G$) information

Description: $M_{12}$
Order: \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \)
Exponent: \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
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)$$M_{12}:C_2$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \)
$\operatorname{Aut}(H)$ $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
$W$$C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \)

Related subgroups

Centralizer:$C_2\times C_6$
Normalizer:$S_3\times A_4$
Normal closure:$M_{12}$
Core:$C_1$
Minimal over-subgroups:$C_3\times A_4$$C_2\times D_6$
Maximal under-subgroups:$C_6$$C_2^2$

Other information

Number of subgroups in this conjugacy class$1320$
Möbius function$0$
Projective image$M_{12}$