Properties

Label 95040.a.47520.b1.a1
Order $ 2 $
Index $ 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 $
Normal No

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Subgroup ($H$) information

Description:$C_2$
Order: \(2\)
Index: \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \)
Exponent: \(2\)
Generators: $\langle(2,4)(5,6)(7,12)(8,10)\rangle$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.

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 \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \)
$\operatorname{Aut}(H)$ $C_1$, of order $1$
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$Q_8:S_4$
Normalizer:$Q_8:S_4$
Normal closure:$M_{12}$
Core:$C_1$
Minimal over-subgroups:$D_5$$C_6$$S_3$$S_3$$S_3$$C_2^2$$C_4$$C_4$$C_2^2$$C_2^2$
Maximal under-subgroups:$C_1$

Other information

Number of subgroups in this conjugacy class$495$
Möbius function$-480$
Projective image$M_{12}$