Normal subgroup $C_3^5:D_6\wr C_2 \trianglelefteq C_3^3.S_3\wr C_2^2$

• Label: 139968.kc.2.B
• Order: $ 2^{5} \cdot 3^{7} $
• Index: $ 2 $
• Normal: Yes

Subgroup ($H$) information

Description:	$C_3^5:D_6\wr C_2$
Order:	\(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Index:	\(2\)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Generators:	$\langle(13,27)(14,25)(15,26)(19,32)(20,33)(21,31), (4,29,18)(5,30,16)(6,28,17) \!\cdots\! \rangle$
Derived length:	$4$

The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial 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:	$C_2$
Order:	\(2\)
Exponent:	\(2\)
Automorphism Group:	$C_1$, of order $1$
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
$\operatorname{Aut}(H)$	$C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
$W$	$C_3^3.S_3\wr C_2^2$, 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:	$C_2$ $C_2$
Minimal over-subgroups:	$C_3^3.S_3\wr C_2^2$
Maximal under-subgroups:	$C_3^4.C_3^3.C_2^4$	$C_3^4:(S_3^2:D_6)$	$C_3^5.S_3^2:C_4$	$C_3^3:S_3^2:S_3^2$	$C_3^5:(C_6^2:C_4)$	$C_3^4:D_6\wr C_2$	$S_3^4:S_3$

Other information

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