Normal subgroup $C_3^4:(D_4\times \GL(2,3)) \trianglelefteq C_3^4:(D_4\times \GL(2,3))$

• Label: 31104.mm.1.a1
• Order: $ 2^{7} \cdot 3^{5} $
• Index: $ 1 $
• Normal: Yes

Subgroup ($H$) information

Description:	$C_3^4:(D_4\times \GL(2,3))$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Index:	$1$
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Generators:	$\langle(2,5,6)(8,9)(10,13)(11,15)(12,14), (7,15,11)(8,13,12)(9,14,10), (1,5)(2,4) \!\cdots\! \rangle$
Derived length:	$5$

The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup. Whether it is monomial has not been computed.

Ambient group ($G$) information

Description:	$C_3^4:(D_4\times \GL(2,3))$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$5$

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

Quotient group ($Q$) structure

Description:	$C_1$
Order:	$1$
Exponent:	$1$
Automorphism Group:	$C_1$, of order $1$
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$0$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3:S_3.C_6^2.C_6.D_4.C_2$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
$\operatorname{Aut}(H)$	$C_3:S_3.C_6^2.C_6.D_4.C_2$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
$W$	$C_3^4:(D_4\times \GL(2,3))$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)

Related subgroups

Centralizer:	$C_1$
Normalizer:	$C_3^4:(D_4\times \GL(2,3))$
Complements:	$C_1$
Maximal under-subgroups:	$S_3^2\times C_3^2:\GL(2,3)$	$(C_3^2\times S_3^2):\GL(2,3)$	$C_3^4:(D_4\times \SL(2,3))$	$C_3^4:(C_4\times \GL(2,3))$	$C_3^3:C_{12}:\GL(2,3)$	$\SOPlus(4,2)^2.C_2$	$C_3^4:(D_4\times D_6)$	$C_2^4.\SL(3,3)$	$\GL(2,3)\times \SOPlus(4,2)$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	$1$
Projective image	$C_3^4:(D_4\times \GL(2,3))$