Properties

Label 12960.bq.8.a1.a1
Order $ 2^{2} \cdot 3^{4} \cdot 5 $
Index $ 2^{3} $
Normal Yes

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Subgroup ($H$) information

Description:$C_5\times C_3^3:A_4$
Order: \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \)
Generators: $\langle(4,9)(5,8)(10,14,13,11,12), (4,7,9), (1,3,6)(4,7,9), (1,2,4)(3,8,7)(5,9,6), (3,6)(7,9), (2,5,8), (10,11,14,12,13)\rangle$ Copy content Toggle raw display
Derived length: $3$

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

Ambient group ($G$) information

Description: $F_5\times S_3\wr C_3$
Order: \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \)
Exponent: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Derived length:$3$

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

Quotient group ($Q$) structure

Description: $C_2\times C_4$
Order: \(8\)\(\medspace = 2^{3} \)
Exponent: \(4\)\(\medspace = 2^{2} \)
Automorphism Group: $D_4$, of order \(8\)\(\medspace = 2^{3} \)
Outer Automorphisms: $D_4$, of order \(8\)\(\medspace = 2^{3} \)
Derived length: $1$

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

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)$$F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \)
$\operatorname{Aut}(H)$ $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
$W$$S_3^3:C_{12}$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)

Related subgroups

Centralizer:$C_5$
Normalizer:$F_5\times S_3\wr C_3$
Complements:$C_2\times C_4$ $C_2\times C_4$ $C_2\times C_4$ $C_2\times C_4$ $C_2\times C_4$ $C_2\times C_4$
Minimal over-subgroups:$D_5\times C_3^3:A_4$$S_3^3:C_{15}$$C_3^3:(D_5\times A_4)$
Maximal under-subgroups:$C_{15}:S_3^2$$C_3^3:C_{15}$$C_3^3:A_4$$C_5\times A_4$

Other information

Möbius function$0$
Projective image$F_5\times S_3\wr C_3$