Properties

Label 432.523.24.a1.a1
Order $ 2 \cdot 3^{2} $
Index $ 2^{3} \cdot 3 $
Normal Yes

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Subgroup ($H$) information

Description:$C_3:S_3$
Order: \(18\)\(\medspace = 2 \cdot 3^{2} \)
Index: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Generators: $b^{3}, d^{2}, c^{2}$ Copy content Toggle raw display
Derived length: $2$

The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.

Ambient group ($G$) information

Description: $C_6^2:D_6$
Order: \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Derived length:$3$

The ambient group is nonabelian, monomial (hence solvable), and rational.

Quotient group ($Q$) structure

Description: $S_4$
Order: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism Group: $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)
Outer Automorphisms: $C_1$, of order $1$
Derived length: $3$

The quotient is nonabelian, monomial (hence solvable), and rational.

Automorphism information

Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$$C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
$\operatorname{Aut}(H)$ $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(4\)\(\medspace = 2^{2} \)
$W$$C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \)

Related subgroups

Centralizer:$C_2^2$
Normalizer:$C_6^2:D_6$
Complements:$S_4$ $S_4$ $S_4$
Minimal over-subgroups:$C_3^2:C_6$$C_6:S_3$$S_3^2$
Maximal under-subgroups:$C_3^2$$S_3$$S_3$

Other information

Möbius function$-12$
Projective image$C_6^2:D_6$