Properties

Label 5184.ff.8.a1
Order $ 2^{3} \cdot 3^{4} $
Index $ 2^{3} $
Normal Yes

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Subgroup ($H$) information

Description:$C_3^4:C_2^3$
Order: \(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Generators: $\langle(1,11,4)(3,9,7), (4,11)(6,10)(7,9)(8,12), (3,9,7), (1,4,11)(2,12,8)(3,7,9)(5,6,10), (5,10,6), (1,2)(3,6,9,5,7,10)(4,8)(11,12), (13,14)(15,16)\rangle$ Copy content Toggle raw display
Derived length: $2$

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

Ambient group ($G$) information

Description: $C_3^4:C_4^2:C_2^2$
Order: \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Derived length:$3$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

Quotient group ($Q$) structure

Description: $C_2^3$
Order: \(8\)\(\medspace = 2^{3} \)
Exponent: \(2\)
Automorphism Group: $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Outer Automorphisms: $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Derived length: $1$

The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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_3^4.C_2^3.C_2^5.C_2^4$
$\operatorname{Aut}(H)$ $(C_3^2\times C_6^2).\GL(2,3)\wr C_2$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \)
$W$$C_3:S_3^3:C_2^2$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)

Related subgroups

Centralizer:$C_2$
Normalizer:$C_3^4:C_4^2:C_2^2$
Minimal over-subgroups:$(C_3^3\times C_6).D_4$$C_6.S_3^3$$C_3^4:(C_2^2\times C_4)$$C_6:S_3^3$$C_2\times C_3^4:D_4$
Maximal under-subgroups:$C_3^3:D_6$$C_3^2:S_3^2$$C_3^2:C_6^2$$C_3^2:S_3^2$$C_6:S_3^2$$C_6:S_3^2$

Other information

Number of subgroups in this autjugacy class$4$
Number of conjugacy classes in this autjugacy class$4$
Möbius function$-8$
Projective image$C_3:S_3^3:C_2^2$