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## G = C13×C22≀C2order 416 = 25·13

### Direct product of C13 and C22≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C13×C22≀C2
 Chief series C1 — C2 — C22 — C2×C26 — C22×C26 — D4×C26 — C13×C22≀C2
 Lower central C1 — C22 — C13×C22≀C2
 Upper central C1 — C2×C26 — C13×C22≀C2

Generators and relations for C13×C22≀C2
G = < a,b,c,d,e,f | a13=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C23, C13, C22⋊C4, C2×D4, C24, C26, C26, C22≀C2, C52, C2×C26, C2×C26, C2×C26, C2×C52, D4×C13, C22×C26, C22×C26, C22×C26, C13×C22⋊C4, D4×C26, C23×C26, C13×C22≀C2
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C22≀C2, C2×C26, D4×C13, C22×C26, D4×C26, C13×C22≀C2

Smallest permutation representation of C13×C22≀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 65)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 27)(25 28)(26 29)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 79)(52 80)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 92)(75 93)(76 94)(77 95)(78 96)
(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 79)(76 80)(77 81)(78 82)
(1 37)(2 38)(3 39)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 53)(23 54)(24 55)(25 56)(26 57)(40 77)(41 78)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 92)
(1 65)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 27)(25 28)(26 29)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 79)(76 80)(77 81)(78 82)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 40)(11 41)(12 42)(13 43)(14 91)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 66)(36 67)(37 68)(38 69)(39 70)(53 100)(54 101)(55 102)(56 103)(57 104)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 13A ··· 13L 26A ··· 26AJ 26AK ··· 26DD 26DE ··· 26DP 52A ··· 52AJ order 1 2 2 2 2 ··· 2 2 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C13 C26 C26 C26 D4 D4×C13 kernel C13×C22≀C2 C13×C22⋊C4 D4×C26 C23×C26 C22≀C2 C22⋊C4 C2×D4 C24 C2×C26 C22 # reps 1 3 3 1 12 36 36 12 6 72

Matrix representation of C13×C22≀C2 in GL4(𝔽53) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 52 0 0 0 0 52 0 0 0 0 52 0 0 0 30 1
,
 1 24 0 0 0 52 0 0 0 0 1 0 0 0 23 52
,
 1 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 24 49 0 0 51 29 0 0 0 0 23 51 0 0 52 30
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,52,30,0,0,0,1],[1,0,0,0,24,52,0,0,0,0,1,23,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[24,51,0,0,49,29,0,0,0,0,23,52,0,0,51,30] >;

C13×C22≀C2 in GAP, Magma, Sage, TeX

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

G:=Group("C13xC2^2wrC2");
// GroupNames label

G:=SmallGroup(416,181);
// by ID

G=gap.SmallGroup(416,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818]);
// Polycyclic

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

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