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G = C41⋊C4order 164 = 22·41

The semidirect product of C41 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C41⋊C4, D41.C2, SmallGroup(164,3)

Series: Derived Chief Lower central Upper central

C1C41 — C41⋊C4
C1C41D41 — C41⋊C4
C41 — C41⋊C4
C1

Generators and relations for C41⋊C4
 G = < a,b | a41=b4=1, bab-1=a9 >

41C2
41C4

Character table of C41⋊C4

 class 124A4B41A41B41C41D41E41F41G41H41I41J
 size 14141414444444444
ρ111111111111111    trivial
ρ211-1-11111111111    linear of order 2
ρ31-1-ii1111111111    linear of order 4
ρ41-1i-i1111111111    linear of order 4
ρ54000ζ413941234118412ζ413841274114413ζ4130412441174111ζ41374136415414ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417    orthogonal faithful
ρ64000ζ4129412641154112ζ413941234118412ζ4125412141204116ζ4130412441174111ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941    orthogonal faithful
ρ74000ζ4130412441174111ζ41374136415414ζ4140413241941ζ413441224119417ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412    orthogonal faithful
ρ84000ζ413841274114413ζ4125412141204116ζ41374136415414ζ413541284113416ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418    orthogonal faithful
ρ94000ζ4140413241941ζ413441224119417ζ4129412641154112ζ413941234118412ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111    orthogonal faithful
ρ104000ζ413541284113416ζ4140413241941ζ413341314110418ζ4129412641154112ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116    orthogonal faithful
ρ114000ζ41374136415414ζ413541284113416ζ413441224119417ζ413341314110418ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413    orthogonal faithful
ρ124000ζ4125412141204116ζ4130412441174111ζ413541284113416ζ4140413241941ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112    orthogonal faithful
ρ134000ζ413341314110418ζ4129412641154112ζ413841274114413ζ4125412141204116ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416    orthogonal faithful
ρ144000ζ413441224119417ζ413341314110418ζ413941234118412ζ413841274114413ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414    orthogonal faithful

Smallest permutation representation of C41⋊C4
On 41 points: primitive
Generators in S41
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41)
(2 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)

G:=sub<Sym(41)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)], [(2,33,41,10),(3,24,40,19),(4,15,39,28),(5,6,38,37),(7,29,36,14),(8,20,35,23),(9,11,34,32),(12,25,31,18),(13,16,30,27),(17,21,26,22)])

C41⋊C4 is a maximal subgroup of   C41⋊C8  C41⋊Dic3
C41⋊C4 is a maximal quotient of   C412C8  C41⋊Dic3

Matrix representation of C41⋊C4 in GL4(𝔽821) generated by

817100
267010
554001
109756401789
,
787474393437
47277249307
652653448756
446396609130
G:=sub<GL(4,GF(821))| [817,267,554,109,1,0,0,756,0,1,0,401,0,0,1,789],[787,47,652,446,474,277,653,396,393,249,448,609,437,307,756,130] >;

C41⋊C4 in GAP, Magma, Sage, TeX

C_{41}\rtimes C_4
% in TeX

G:=Group("C41:C4");
// GroupNames label

G:=SmallGroup(164,3);
// by ID

G=gap.SmallGroup(164,3);
# by ID

G:=PCGroup([3,-2,-2,-41,6,1154,725]);
// Polycyclic

G:=Group<a,b|a^41=b^4=1,b*a*b^-1=a^9>;
// generators/relations

Export

Subgroup lattice of C41⋊C4 in TeX
Character table of C41⋊C4 in TeX

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