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## G = C13×C4≀C2order 416 = 25·13

### Direct product of C13 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C52 — C13×M4(2) — C13×C4≀C2
 Lower central C1 — C2 — C4 — C13×C4≀C2
 Upper central C1 — C52 — C2×C52 — C13×C4≀C2

Generators and relations for C13×C4≀C2
G = < a,b,c,d | a13=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Smallest permutation representation of C13×C4≀C2
On 104 points
Generators in S104
(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)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 55 80 93)(2 56 81 94)(3 57 82 95)(4 58 83 96)(5 59 84 97)(6 60 85 98)(7 61 86 99)(8 62 87 100)(9 63 88 101)(10 64 89 102)(11 65 90 103)(12 53 91 104)(13 54 79 92)(14 46 66 32)(15 47 67 33)(16 48 68 34)(17 49 69 35)(18 50 70 36)(19 51 71 37)(20 52 72 38)(21 40 73 39)(22 41 74 27)(23 42 75 28)(24 43 76 29)(25 44 77 30)(26 45 78 31)
(1 78)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 79)(26 80)(27 102)(28 103)(29 104)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 63)(41 64)(42 65)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)
(1 93 80 55)(2 94 81 56)(3 95 82 57)(4 96 83 58)(5 97 84 59)(6 98 85 60)(7 99 86 61)(8 100 87 62)(9 101 88 63)(10 102 89 64)(11 103 90 65)(12 104 91 53)(13 92 79 54)

G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,55,80,93)(2,56,81,94)(3,57,82,95)(4,58,83,96)(5,59,84,97)(6,60,85,98)(7,61,86,99)(8,62,87,100)(9,63,88,101)(10,64,89,102)(11,65,90,103)(12,53,91,104)(13,54,79,92)(14,46,66,32)(15,47,67,33)(16,48,68,34)(17,49,69,35)(18,50,70,36)(19,51,71,37)(20,52,72,38)(21,40,73,39)(22,41,74,27)(23,42,75,28)(24,43,76,29)(25,44,77,30)(26,45,78,31), (1,78)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,79)(26,80)(27,102)(28,103)(29,104)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,63)(41,64)(42,65)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62), (1,93,80,55)(2,94,81,56)(3,95,82,57)(4,96,83,58)(5,97,84,59)(6,98,85,60)(7,99,86,61)(8,100,87,62)(9,101,88,63)(10,102,89,64)(11,103,90,65)(12,104,91,53)(13,92,79,54)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,55,80,93)(2,56,81,94)(3,57,82,95)(4,58,83,96)(5,59,84,97)(6,60,85,98)(7,61,86,99)(8,62,87,100)(9,63,88,101)(10,64,89,102)(11,65,90,103)(12,53,91,104)(13,54,79,92)(14,46,66,32)(15,47,67,33)(16,48,68,34)(17,49,69,35)(18,50,70,36)(19,51,71,37)(20,52,72,38)(21,40,73,39)(22,41,74,27)(23,42,75,28)(24,43,76,29)(25,44,77,30)(26,45,78,31), (1,78)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,79)(26,80)(27,102)(28,103)(29,104)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,63)(41,64)(42,65)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62), (1,93,80,55)(2,94,81,56)(3,95,82,57)(4,96,83,58)(5,97,84,59)(6,98,85,60)(7,99,86,61)(8,100,87,62)(9,101,88,63)(10,102,89,64)(11,103,90,65)(12,104,91,53)(13,92,79,54) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,55,80,93),(2,56,81,94),(3,57,82,95),(4,58,83,96),(5,59,84,97),(6,60,85,98),(7,61,86,99),(8,62,87,100),(9,63,88,101),(10,64,89,102),(11,65,90,103),(12,53,91,104),(13,54,79,92),(14,46,66,32),(15,47,67,33),(16,48,68,34),(17,49,69,35),(18,50,70,36),(19,51,71,37),(20,52,72,38),(21,40,73,39),(22,41,74,27),(23,42,75,28),(24,43,76,29),(25,44,77,30),(26,45,78,31)], [(1,78),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,79),(26,80),(27,102),(28,103),(29,104),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,63),(41,64),(42,65),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62)], [(1,93,80,55),(2,94,81,56),(3,95,82,57),(4,96,83,58),(5,97,84,59),(6,98,85,60),(7,99,86,61),(8,100,87,62),(9,101,88,63),(10,102,89,64),(11,103,90,65),(12,104,91,53),(13,92,79,54)])

182 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4G 4H 8A 8B 13A ··· 13L 26A ··· 26L 26M ··· 26X 26Y ··· 26AJ 52A ··· 52X 52Y ··· 52CF 52CG ··· 52CR 104A ··· 104X order 1 2 2 2 4 4 4 ··· 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 4 1 1 2 ··· 2 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C13 C26 C26 C26 C52 C52 D4 D4 C4≀C2 D4×C13 D4×C13 C13×C4≀C2 kernel C13×C4≀C2 C4×C52 C13×M4(2) C13×C4○D4 D4×C13 Q8×C13 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C52 C2×C26 C13 C4 C22 C1 # reps 1 1 1 1 2 2 12 12 12 12 24 24 1 1 4 12 12 48

Matrix representation of C13×C4≀C2 in GL2(𝔽313) generated by

 277 0 0 277
,
 288 0 0 25
,
 0 25 288 0
,
 25 0 0 1
G:=sub<GL(2,GF(313))| [277,0,0,277],[288,0,0,25],[0,288,25,0],[25,0,0,1] >;

C13×C4≀C2 in GAP, Magma, Sage, TeX

C_{13}\times C_4\wr C_2
% in TeX

G:=Group("C13xC4wrC2");
// GroupNames label

G:=SmallGroup(416,54);
// by ID

G=gap.SmallGroup(416,54);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117,88]);
// Polycyclic

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

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