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## G = C13×2+ 1+4order 416 = 25·13

### Direct product of C13 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×2+ 1+4
 Chief series C1 — C2 — C26 — C2×C26 — D4×C13 — D4×C26 — C13×2+ 1+4
 Lower central C1 — C2 — C13×2+ 1+4
 Upper central C1 — C26 — C13×2+ 1+4

Generators and relations for C13×2+ 1+4
G = < a,b,c,d,e | a13=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2 [×9], C4 [×6], C22 [×9], C22 [×6], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C13, C2×D4 [×9], C4○D4 [×6], C26, C26 [×9], 2+ 1+4, C52 [×6], C2×C26 [×9], C2×C26 [×6], C2×C52 [×9], D4×C13 [×18], Q8×C13 [×2], C22×C26 [×6], D4×C26 [×9], C13×C4○D4 [×6], C13×2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C13, C24, C26 [×15], 2+ 1+4, C2×C26 [×35], C22×C26 [×15], C23×C26, C13×2+ 1+4

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

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

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

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

221 conjugacy classes

 class 1 2A 2B ··· 2J 4A ··· 4F 13A ··· 13L 26A ··· 26L 26M ··· 26DP 52A ··· 52BT order 1 2 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

221 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C13 C26 C26 2+ 1+4 C13×2+ 1+4 kernel C13×2+ 1+4 D4×C26 C13×C4○D4 2+ 1+4 C2×D4 C4○D4 C13 C1 # reps 1 9 6 12 108 72 1 12

Matrix representation of C13×2+ 1+4 in GL4(𝔽53) generated by

 24 0 0 0 0 24 0 0 0 0 24 0 0 0 0 24
,
 1 8 0 2 0 7 51 0 0 25 46 0 52 21 8 52
,
 0 1 0 0 1 0 0 0 4 4 0 1 49 49 1 0
,
 52 45 0 51 0 7 51 0 0 25 46 0 1 29 8 1
,
 0 7 51 0 52 45 0 51 49 25 0 46 4 28 1 8
G:=sub<GL(4,GF(53))| [24,0,0,0,0,24,0,0,0,0,24,0,0,0,0,24],[1,0,0,52,8,7,25,21,0,51,46,8,2,0,0,52],[0,1,4,49,1,0,4,49,0,0,0,1,0,0,1,0],[52,0,0,1,45,7,25,29,0,51,46,8,51,0,0,1],[0,52,49,4,7,45,25,28,51,0,0,1,0,51,46,8] >;

C13×2+ 1+4 in GAP, Magma, Sage, TeX

C_{13}\times 2_+^{1+4}
% in TeX

G:=Group("C13xES+(2,2)");
// GroupNames label

G:=SmallGroup(416,231);
// by ID

G=gap.SmallGroup(416,231);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,1916,5187]);
// Polycyclic

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

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