Normal subgroup $C_3^4.S_3^3 \trianglelefteq C_3^3.S_3\wr C_2^2$

• Label: 139968.kc.8.B
• Order: $ 2^{3} \cdot 3^{7} $
• Index: $ 2^{3} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(17496\)\(\medspace = 2^{3} \cdot 3^{7} \)
Index:	\(8\)\(\medspace = 2^{3} \)
Exponent:	not computed
Generators:	$\langle(13,27)(14,25)(15,26)(19,32)(20,33)(21,31), (1,3)(5,6)(8,9)(10,11)(13,25) \!\cdots\! \rangle$
Derived length:	not computed

The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description:	$C_3^3.S_3\wr C_2^2$
Order:	\(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Derived length:	$4$

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

Quotient group ($Q$) structure

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

The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
$\operatorname{Aut}(H)$	not computed
$W$	$C_3^3.S_3\wr C_2^2$, of order \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)

Related subgroups

Centralizer:	$C_1$
Normalizer:	$C_3^3.S_3\wr C_2^2$
Complements:	$D_4$ $D_4$ $D_4$ $D_4$
Minimal over-subgroups:	$C_3^2.C_3^5.C_2^3.C_2$	$C_3.C_3^5.D_6.C_2^2$	$C_3^4.S_3^3.C_2$
Maximal under-subgroups:	$C_3^4.C_3^3.C_2^2$	$C_3^5.(C_6\times S_3)$	$C_3^5.C_3.D_6$	$C_3.C_3^5.C_2^3$	$C_3^3.S_3^3$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	$C_3^3.S_3\wr C_2^2$