Properties

Label 4032.fk.4.a1.b1
Order $ 2^{4} \cdot 3^{2} \cdot 7 $
Index $ 2^{2} $
Normal Yes

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Subgroup ($H$) information

Description:$C_7:C_6\times S_4$
Order: \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Index: \(4\)\(\medspace = 2^{2} \)
Exponent: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Generators: $ac^{49}e, de, b^{2}, c^{28}e, c^{42}, e, c^{12}$ Copy content Toggle raw display
Derived length: $3$

The subgroup is normal, a semidirect factor, nonabelian, and monomial (hence solvable).

Ambient group ($G$) information

Description: $C_{28}:(C_6\times S_4)$
Order: \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Exponent: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Derived length:$3$

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

Quotient group ($Q$) structure

Description: $C_2^2$
Order: \(4\)\(\medspace = 2^{2} \)
Exponent: \(2\)
Automorphism Group: $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Outer Automorphisms: $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Derived length: $1$

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

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_{14}\times A_4).C_6.C_2^4$
$\operatorname{Aut}(H)$ $C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
$\operatorname{res}(S)$$C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(4\)\(\medspace = 2^{2} \)
$W$$C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)

Related subgroups

Centralizer:$C_2$
Normalizer:$C_{28}:(C_6\times S_4)$
Complements:$C_2^2$ $C_2^2$ $C_2^2$ $C_2^2$
Minimal over-subgroups:$C_2\times S_4\times F_7$$(C_{14}\times S_4):C_6$$C_{28}:(C_3\times S_4)$
Maximal under-subgroups:$C_7:C_6\times A_4$$C_7:C_3\times S_4$$C_{14}\times S_4$$C_2\times C_{28}:C_6$$C_{42}:C_6$$C_6\times S_4$
Autjugate subgroups:4032.fk.4.a1.a1

Other information

Möbius function$2$
Projective image$C_2\times S_4\times F_7$