Properties

Label 672.1172.1.a1
Order $ 2^{5} \cdot 3 \cdot 7 $
Index $ 1 $
Normal Yes

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Subgroup ($H$) information

Description:$D_{42}:C_2^3$
Order: \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Index: $1$
Exponent: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Generators: $a, d^{2}, c^{7}, d^{3}, b, c^{4}, c^{14}$ Copy content Toggle raw display
Derived length: $2$

The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.

Ambient group ($G$) information

Description: $D_{42}:C_2^3$
Order: \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Derived length:$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Quotient group ($Q$) structure

Description: $C_1$
Order: $1$
Exponent: $1$
Automorphism Group: $C_1$, of order $1$
Outer Automorphisms: $C_1$, of order $1$
Derived length: $0$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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)$$C_2^4.C_2^4.C_7.C_3^3.C_2^3$
$\operatorname{Aut}(H)$ $C_2^4.C_2^4.C_7.C_3^3.C_2^3$
$W$$S_3\times D_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)

Related subgroups

Centralizer:$C_2^3$
Normalizer:$D_{42}:C_2^3$
Complements:$C_1$
Maximal under-subgroups:$C_6:D_{28}$$C_{42}:C_2^3$$C_2^2\times D_{42}$$C_{42}.C_2^3$$C_2^2\times D_{28}$$C_2^3:D_6$

Other information

Number of conjugacy classes in this autjugacy class$1$
Möbius function$1$
Projective image$S_3\times D_7$