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## G = D5×C28order 280 = 23·5·7

### Direct product of C28 and D5

Aliases: D5×C28, C202C14, C1406C2, Dic52C14, D10.2C14, C14.14D10, C70.19C22, C359(C2×C4), C52(C2×C28), C2.1(D5×C14), C10.2(C2×C14), (C7×Dic5)⋊5C2, (D5×C14).4C2, SmallGroup(280,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C28
 Chief series C1 — C5 — C10 — C70 — D5×C14 — D5×C28
 Lower central C5 — D5×C28
 Upper central C1 — C28

Generators and relations for D5×C28
G = < a,b,c | a28=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C28
On 140 points
Generators in S140
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 107 51 116 80)(2 108 52 117 81)(3 109 53 118 82)(4 110 54 119 83)(5 111 55 120 84)(6 112 56 121 57)(7 85 29 122 58)(8 86 30 123 59)(9 87 31 124 60)(10 88 32 125 61)(11 89 33 126 62)(12 90 34 127 63)(13 91 35 128 64)(14 92 36 129 65)(15 93 37 130 66)(16 94 38 131 67)(17 95 39 132 68)(18 96 40 133 69)(19 97 41 134 70)(20 98 42 135 71)(21 99 43 136 72)(22 100 44 137 73)(23 101 45 138 74)(24 102 46 139 75)(25 103 47 140 76)(26 104 48 113 77)(27 105 49 114 78)(28 106 50 115 79)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 113)(105 114)(106 115)(107 116)(108 117)(109 118)(110 119)(111 120)(112 121)

G:=sub<Sym(140)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,107,51,116,80)(2,108,52,117,81)(3,109,53,118,82)(4,110,54,119,83)(5,111,55,120,84)(6,112,56,121,57)(7,85,29,122,58)(8,86,30,123,59)(9,87,31,124,60)(10,88,32,125,61)(11,89,33,126,62)(12,90,34,127,63)(13,91,35,128,64)(14,92,36,129,65)(15,93,37,130,66)(16,94,38,131,67)(17,95,39,132,68)(18,96,40,133,69)(19,97,41,134,70)(20,98,42,135,71)(21,99,43,136,72)(22,100,44,137,73)(23,101,45,138,74)(24,102,46,139,75)(25,103,47,140,76)(26,104,48,113,77)(27,105,49,114,78)(28,106,50,115,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(103,140)(104,113)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121)>;

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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,107,51,116,80)(2,108,52,117,81)(3,109,53,118,82)(4,110,54,119,83)(5,111,55,120,84)(6,112,56,121,57)(7,85,29,122,58)(8,86,30,123,59)(9,87,31,124,60)(10,88,32,125,61)(11,89,33,126,62)(12,90,34,127,63)(13,91,35,128,64)(14,92,36,129,65)(15,93,37,130,66)(16,94,38,131,67)(17,95,39,132,68)(18,96,40,133,69)(19,97,41,134,70)(20,98,42,135,71)(21,99,43,136,72)(22,100,44,137,73)(23,101,45,138,74)(24,102,46,139,75)(25,103,47,140,76)(26,104,48,113,77)(27,105,49,114,78)(28,106,50,115,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,138)(102,139)(103,140)(104,113)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121) );

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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,107,51,116,80),(2,108,52,117,81),(3,109,53,118,82),(4,110,54,119,83),(5,111,55,120,84),(6,112,56,121,57),(7,85,29,122,58),(8,86,30,123,59),(9,87,31,124,60),(10,88,32,125,61),(11,89,33,126,62),(12,90,34,127,63),(13,91,35,128,64),(14,92,36,129,65),(15,93,37,130,66),(16,94,38,131,67),(17,95,39,132,68),(18,96,40,133,69),(19,97,41,134,70),(20,98,42,135,71),(21,99,43,136,72),(22,100,44,137,73),(23,101,45,138,74),(24,102,46,139,75),(25,103,47,140,76),(26,104,48,113,77),(27,105,49,114,78),(28,106,50,115,79)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,129),(93,130),(94,131),(95,132),(96,133),(97,134),(98,135),(99,136),(100,137),(101,138),(102,139),(103,140),(104,113),(105,114),(106,115),(107,116),(108,117),(109,118),(110,119),(111,120),(112,121)])

112 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 7A ··· 7F 10A 10B 14A ··· 14F 14G ··· 14R 20A 20B 20C 20D 28A ··· 28L 28M ··· 28X 35A ··· 35L 70A ··· 70L 140A ··· 140X order 1 2 2 2 4 4 4 4 5 5 7 ··· 7 10 10 14 ··· 14 14 ··· 14 20 20 20 20 28 ··· 28 28 ··· 28 35 ··· 35 70 ··· 70 140 ··· 140 size 1 1 5 5 1 1 5 5 2 2 1 ··· 1 2 2 1 ··· 1 5 ··· 5 2 2 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D5 D10 C4×D5 C7×D5 D5×C14 D5×C28 kernel D5×C28 C7×Dic5 C140 D5×C14 C7×D5 C4×D5 Dic5 C20 D10 D5 C28 C14 C7 C4 C2 C1 # reps 1 1 1 1 4 6 6 6 6 24 2 2 4 12 12 24

Matrix representation of D5×C28 in GL2(𝔽29) generated by

 3 0 0 3
,
 0 25 22 5
,
 24 20 22 5
G:=sub<GL(2,GF(29))| [3,0,0,3],[0,22,25,5],[24,22,20,5] >;

D5×C28 in GAP, Magma, Sage, TeX

D_5\times C_{28}
% in TeX

G:=Group("D5xC28");
// GroupNames label

G:=SmallGroup(280,20);
// by ID

G=gap.SmallGroup(280,20);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-5,146,5604]);
// Polycyclic

G:=Group<a,b,c|a^28=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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