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## G = C4×C13⋊C4order 208 = 24·13

### Direct product of C4 and C13⋊C4

Aliases: C4×C13⋊C4, C13⋊C42, C522C4, Dic132C4, D26.4C22, D13.(C2×C4), C26.3(C2×C4), (C4×D13).6C2, C2.2(C2×C13⋊C4), (C2×C13⋊C4).2C2, SmallGroup(208,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C4×C13⋊C4
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — C4×C13⋊C4
 Lower central C13 — C4×C13⋊C4
 Upper central C1 — C4

Generators and relations for C4×C13⋊C4
G = < a,b,c | a4=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C4×C13⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 13A 13B 13C 26A 26B 26C 52A 52B 52C 52D 52E 52F size 1 1 13 13 1 1 13 13 13 13 13 13 13 13 13 13 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 -i i -1 -i i i -i -i i -1 1 1 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ6 1 -1 1 -1 -i i i i -i -1 1 -1 1 -i i -i 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ7 1 1 -1 -1 -1 -1 i 1 1 i i -i -i -i -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 i -i i -i i 1 -1 1 -1 -i i -i 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ9 1 -1 -1 1 i -i -1 i -i -i i i -i -1 1 1 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ10 1 1 -1 -1 1 1 i -1 -1 -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ11 1 -1 1 -1 -i i -i i -i 1 -1 1 -1 i -i i 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ12 1 1 -1 -1 -1 -1 -i 1 1 -i -i i i i i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ13 1 -1 -1 1 -i i 1 -i i -i i i -i 1 -1 -1 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ14 1 1 -1 -1 1 1 -i -1 -1 i i -i -i i i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ15 1 -1 1 -1 i -i -i -i i -1 1 -1 1 i -i i 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ16 1 -1 -1 1 i -i 1 i -i i -i -i i 1 -1 -1 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ17 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ18 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ19 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4 ρ20 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ21 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ22 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ23 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 complex faithful ρ24 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 complex faithful ρ25 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 complex faithful ρ26 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 complex faithful ρ27 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 complex faithful ρ28 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ4ζ1312+ζ4ζ138+ζ4ζ135+ζ4ζ13 ζ4ζ1311+ζ4ζ1310+ζ4ζ133+ζ4ζ132 ζ43ζ1312+ζ43ζ138+ζ43ζ135+ζ43ζ13 ζ43ζ1311+ζ43ζ1310+ζ43ζ133+ζ43ζ132 ζ43ζ139+ζ43ζ137+ζ43ζ136+ζ43ζ134 ζ4ζ139+ζ4ζ137+ζ4ζ136+ζ4ζ134 complex faithful

Smallest permutation representation of C4×C13⋊C4
On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(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 42 43 44 45 46 47 48 49 50 51 52)
(1 40 14 27)(2 48 26 32)(3 43 25 37)(4 51 24 29)(5 46 23 34)(6 41 22 39)(7 49 21 31)(8 44 20 36)(9 52 19 28)(10 47 18 33)(11 42 17 38)(12 50 16 30)(13 45 15 35)

G:=sub<Sym(52)| (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,40,14,27)(2,48,26,32)(3,43,25,37)(4,51,24,29)(5,46,23,34)(6,41,22,39)(7,49,21,31)(8,44,20,36)(9,52,19,28)(10,47,18,33)(11,42,17,38)(12,50,16,30)(13,45,15,35)>;

G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,40,14,27)(2,48,26,32)(3,43,25,37)(4,51,24,29)(5,46,23,34)(6,41,22,39)(7,49,21,31)(8,44,20,36)(9,52,19,28)(10,47,18,33)(11,42,17,38)(12,50,16,30)(13,45,15,35) );

G=PermutationGroup([[(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(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),(40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,40,14,27),(2,48,26,32),(3,43,25,37),(4,51,24,29),(5,46,23,34),(6,41,22,39),(7,49,21,31),(8,44,20,36),(9,52,19,28),(10,47,18,33),(11,42,17,38),(12,50,16,30),(13,45,15,35)]])

C4×C13⋊C4 is a maximal subgroup of   C104⋊C4  Dic26⋊C4  D52⋊C4  D26.C23
C4×C13⋊C4 is a maximal quotient of   C104⋊C4  C26.C42  D26.Q8

Matrix representation of C4×C13⋊C4 in GL4(𝔽5) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 2 0 0 3 2 0 3 0 0 0 3 3 1 1 1 2
,
 4 1 2 0 0 2 0 4 0 4 4 0 0 3 4 0
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[2,2,0,1,0,0,0,1,0,3,3,1,3,0,3,2],[4,0,0,0,1,2,4,3,2,0,4,4,0,4,0,0] >;

C4×C13⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes C_4
% in TeX

G:=Group("C4xC13:C4");
// GroupNames label

G:=SmallGroup(208,30);
// by ID

G=gap.SmallGroup(208,30);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,3204,1214]);
// Polycyclic

G:=Group<a,b,c|a^4=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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