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## G = C13⋊A4order 156 = 22·3·13

### The semidirect product of C13 and A4 acting via A4/C22=C3

Aliases: C13⋊A4, (C2×C26)⋊2C3, C22⋊(C13⋊C3), SmallGroup(156,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C13⋊A4
 Chief series C1 — C13 — C2×C26 — C13⋊A4
 Lower central C2×C26 — C13⋊A4
 Upper central C1

Generators and relations for C13⋊A4
G = < a,b,c,d | a13=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a9, dbd-1=bc=cb, dcd-1=b >

Character table of C13⋊A4

 class 1 2 3A 3B 13A 13B 13C 13D 26A 26B 26C 26D 26E 26F 26G 26H 26I 26J 26K 26L size 1 3 52 52 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 -1 0 0 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ5 3 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312-ζ1310-ζ134 -ζ139+ζ133-ζ13 -ζ1311-ζ138+ζ137 ζ1311-ζ138-ζ137 -ζ136-ζ135+ζ132 ζ136-ζ135-ζ132 -ζ1312+ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ136+ζ135-ζ132 ζ139-ζ133-ζ13 -ζ1312-ζ1310+ζ134 -ζ1311+ζ138-ζ137 complex faithful ρ6 3 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136-ζ135-ζ132 -ζ1311+ζ138-ζ137 -ζ1312+ζ1310-ζ134 ζ1312-ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ139+ζ133-ζ13 -ζ136+ζ135-ζ132 -ζ1311-ζ138+ζ137 ζ139-ζ133-ζ13 ζ1311-ζ138-ζ137 -ζ136-ζ135+ζ132 -ζ1312-ζ1310+ζ134 complex faithful ρ7 3 3 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ8 3 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ1312-ζ1310+ζ134 -ζ139-ζ133+ζ13 ζ1311-ζ138-ζ137 -ζ1311+ζ138-ζ137 -ζ136+ζ135-ζ132 -ζ136-ζ135+ζ132 ζ1312-ζ1310-ζ134 ζ139-ζ133-ζ13 ζ136-ζ135-ζ132 -ζ139+ζ133-ζ13 -ζ1312+ζ1310-ζ134 -ζ1311-ζ138+ζ137 complex faithful ρ9 3 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311-ζ138-ζ137 ζ136-ζ135-ζ132 -ζ139-ζ133+ζ13 ζ139-ζ133-ζ13 -ζ1312-ζ1310+ζ134 ζ1312-ζ1310-ζ134 -ζ1311-ζ138+ζ137 -ζ136-ζ135+ζ132 -ζ1312+ζ1310-ζ134 -ζ136+ζ135-ζ132 -ζ1311+ζ138-ζ137 -ζ139+ζ133-ζ13 complex faithful ρ10 3 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ1311+ζ138-ζ137 -ζ136-ζ135+ζ132 ζ139-ζ133-ζ13 -ζ139+ζ133-ζ13 -ζ1312+ζ1310-ζ134 -ζ1312-ζ1310+ζ134 ζ1311-ζ138-ζ137 -ζ136+ζ135-ζ132 ζ1312-ζ1310-ζ134 ζ136-ζ135-ζ132 -ζ1311-ζ138+ζ137 -ζ139-ζ133+ζ13 complex faithful ρ11 3 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ1311-ζ138+ζ137 -ζ136+ζ135-ζ132 -ζ139+ζ133-ζ13 -ζ139-ζ133+ζ13 ζ1312-ζ1310-ζ134 -ζ1312+ζ1310-ζ134 -ζ1311+ζ138-ζ137 ζ136-ζ135-ζ132 -ζ1312-ζ1310+ζ134 -ζ136-ζ135+ζ132 ζ1311-ζ138-ζ137 ζ139-ζ133-ζ13 complex faithful ρ12 3 3 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ13 3 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ136+ζ135-ζ132 ζ1311-ζ138-ζ137 -ζ1312-ζ1310+ζ134 -ζ1312+ζ1310-ζ134 -ζ139+ζ133-ζ13 ζ139-ζ133-ζ13 -ζ136-ζ135+ζ132 -ζ1311+ζ138-ζ137 -ζ139-ζ133+ζ13 -ζ1311-ζ138+ζ137 ζ136-ζ135-ζ132 ζ1312-ζ1310-ζ134 complex faithful ρ14 3 3 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ15 3 3 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ16 3 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ139-ζ133+ζ13 -ζ1312+ζ1310-ζ134 ζ136-ζ135-ζ132 -ζ136-ζ135+ζ132 ζ1311-ζ138-ζ137 -ζ1311-ζ138+ζ137 -ζ139+ζ133-ζ13 ζ1312-ζ1310-ζ134 -ζ1311+ζ138-ζ137 -ζ1312-ζ1310+ζ134 ζ139-ζ133-ζ13 -ζ136+ζ135-ζ132 complex faithful ρ17 3 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139-ζ133-ζ13 ζ1312-ζ1310-ζ134 -ζ136-ζ135+ζ132 -ζ136+ζ135-ζ132 -ζ1311+ζ138-ζ137 ζ1311-ζ138-ζ137 -ζ139-ζ133+ζ13 -ζ1312-ζ1310+ζ134 -ζ1311-ζ138+ζ137 -ζ1312+ζ1310-ζ134 -ζ139+ζ133-ζ13 ζ136-ζ135-ζ132 complex faithful ρ18 3 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ136-ζ135+ζ132 -ζ1311-ζ138+ζ137 ζ1312-ζ1310-ζ134 -ζ1312-ζ1310+ζ134 ζ139-ζ133-ζ13 -ζ139-ζ133+ζ13 ζ136-ζ135-ζ132 ζ1311-ζ138-ζ137 -ζ139+ζ133-ζ13 -ζ1311+ζ138-ζ137 -ζ136+ζ135-ζ132 -ζ1312+ζ1310-ζ134 complex faithful ρ19 3 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ139+ζ133-ζ13 -ζ1312-ζ1310+ζ134 -ζ136+ζ135-ζ132 ζ136-ζ135-ζ132 -ζ1311-ζ138+ζ137 -ζ1311+ζ138-ζ137 ζ139-ζ133-ζ13 -ζ1312+ζ1310-ζ134 ζ1311-ζ138-ζ137 ζ1312-ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ136-ζ135+ζ132 complex faithful ρ20 3 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ1312+ζ1310-ζ134 ζ139-ζ133-ζ13 -ζ1311+ζ138-ζ137 -ζ1311-ζ138+ζ137 ζ136-ζ135-ζ132 -ζ136+ζ135-ζ132 -ζ1312-ζ1310+ζ134 -ζ139+ζ133-ζ13 -ζ136-ζ135+ζ132 -ζ139-ζ133+ζ13 ζ1312-ζ1310-ζ134 ζ1311-ζ138-ζ137 complex faithful

Smallest permutation representation of C13⋊A4
On 52 points
Generators in S52
(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 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 27)(13 28)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 14)(11 15)(12 16)(13 17)(27 51)(28 52)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(14 41 32)(15 44 28)(16 47 37)(17 50 33)(18 40 29)(19 43 38)(20 46 34)(21 49 30)(22 52 39)(23 42 35)(24 45 31)(25 48 27)(26 51 36)

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

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

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

C13⋊A4 is a maximal subgroup of   D13⋊A4  A4×C13⋊C3
C13⋊A4 is a maximal quotient of   C26.A4  C39.A4

Matrix representation of C13⋊A4 in GL3(𝔽79) generated by

 0 1 0 0 0 1 1 10 25
,
 15 38 51 51 51 49 49 67 12
,
 10 24 73 73 29 32 32 77 39
,
 1 0 0 24 68 54 3 55 10
G:=sub<GL(3,GF(79))| [0,0,1,1,0,10,0,1,25],[15,51,49,38,51,67,51,49,12],[10,73,32,24,29,77,73,32,39],[1,24,3,0,68,55,0,54,10] >;

C13⋊A4 in GAP, Magma, Sage, TeX

C_{13}\rtimes A_4
% in TeX

G:=Group("C13:A4");
// GroupNames label

G:=SmallGroup(156,14);
// by ID

G=gap.SmallGroup(156,14);
# by ID

G:=PCGroup([4,-3,-2,2,-13,49,110,579]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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