Properties

Label 1344.9757.6.h1.b1
Order $ 2^{5} \cdot 7 $
Index $ 2 \cdot 3 $
Normal No

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Subgroup ($H$) information

Description:$C_8.D_{14}$
Order: \(224\)\(\medspace = 2^{5} \cdot 7 \)
Index: \(6\)\(\medspace = 2 \cdot 3 \)
Exponent: \(56\)\(\medspace = 2^{3} \cdot 7 \)
Generators: $ad^{45}, d^{12}, c^{3}d^{7}, c^{2}d^{42}, b, d^{42}$ Copy content Toggle raw display
Derived length: $2$

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

Ambient group ($G$) information

Description: $C_{84}.C_2^4$
Order: \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent: \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Derived length:$2$

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

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_{42}.(C_2^5\times C_6).C_2^2$
$\operatorname{Aut}(H)$ $C_2^2\times D_4\times F_7$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
$\card{W}$\(224\)\(\medspace = 2^{5} \cdot 7 \)

Related subgroups

Centralizer:$C_2$
Normalizer:$Q_{16}.D_{14}$
Normal closure:$D_{12}.D_{14}$
Core:$D_{28}:C_2$
Minimal over-subgroups:$D_{12}.D_{14}$$Q_{16}.D_{14}$
Maximal under-subgroups:$D_{28}:C_2$$D_4:D_7$$C_7:D_8$$C_7\times \SD_{16}$$C_8\times D_7$$C_{56}:C_2$$C_7:Q_{16}$$D_8:C_2$
Autjugate subgroups:1344.9757.6.h1.a1

Other information

Number of subgroups in this conjugacy class$3$
Möbius function not computed
Projective image not computed