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## G = C22×C13⋊C4order 208 = 24·13

### Direct product of C22 and C13⋊C4

Aliases: C22×C13⋊C4, D263C4, D13.C23, D26.7C22, C26⋊(C2×C4), D13⋊(C2×C4), C13⋊(C22×C4), (C2×C26)⋊2C4, (C22×D13).3C2, SmallGroup(208,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×C13⋊C4
 Chief series C1 — C13 — D13 — C13⋊C4 — C2×C13⋊C4 — C22×C13⋊C4
 Lower central C13 — C22×C13⋊C4
 Upper central C1 — C22

Generators and relations for C22×C13⋊C4
G = < a,b,c,d | a2=b2=c13=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 318 in 54 conjugacy classes, 32 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C23, C13, C22×C4, D13, D13, C26, C13⋊C4, D26, C2×C26, C2×C13⋊C4, C22×D13, C22×C13⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4

Character table of C22×C13⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 13A 13B 13C 26A 26B 26C 26D 26E 26F 26G 26H 26I size 1 1 1 1 13 13 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 -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 ρ6 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 ρ7 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 ρ8 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 ρ9 1 -1 1 -1 -1 1 -1 1 -i -i i i -i -i i i 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ10 1 -1 -1 1 1 1 -1 -1 -i 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 -1 1 -1 1 i i -i -i i i -i -i 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ12 1 -1 -1 1 1 1 -1 -1 i -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 1 -1 -1 1 i i -i i -i -i i -i 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 i -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 1 -1 -1 1 -i -i i -i i i -i i 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 -i i -i i -i i -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ17 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ18 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C2×C13⋊C4 ρ20 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C2×C13⋊C4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1312-ζ138-ζ135-ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 orthogonal lifted from C2×C13⋊C4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C2×C13⋊C4 ρ24 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ139-ζ137-ζ136-ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1311-ζ1310-ζ133-ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ139-ζ137-ζ136-ζ134 orthogonal lifted from C2×C13⋊C4 ρ27 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ28 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ1312-ζ138-ζ135-ζ13 orthogonal lifted from C2×C13⋊C4

Smallest permutation representation of C22×C13⋊C4
On 52 points
Generators in S52
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 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 14)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)(27 40)(28 48 39 45)(29 43 38 50)(30 51 37 42)(31 46 36 47)(32 41 35 52)(33 49 34 44)

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

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

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

C22×C13⋊C4 is a maximal subgroup of   D26.Q8
C22×C13⋊C4 is a maximal quotient of   D13⋊M4(2)  D26.C23  Dic26.C4  D52.C4

Matrix representation of C22×C13⋊C4 in GL5(𝔽53)

 1 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 52
,
 52 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 15 38 21 52 0 16 38 21 52 0 15 39 21 52 0 15 38 22 52
,
 30 0 0 0 0 0 21 38 15 32 0 1 39 21 51 0 0 52 0 0 0 21 39 1 46

G:=sub<GL(5,GF(53))| [1,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52],[52,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,15,16,15,15,0,38,38,39,38,0,21,21,21,22,0,52,52,52,52],[30,0,0,0,0,0,21,1,0,21,0,38,39,52,39,0,15,21,0,1,0,32,51,0,46] >;

C22×C13⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_{13}\rtimes C_4
% in TeX

G:=Group("C2^2xC13:C4");
// GroupNames label

G:=SmallGroup(208,49);
// by ID

G=gap.SmallGroup(208,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,3204,619]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^13=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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