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## G = D8×C19order 304 = 24·19

### Direct product of C19 and D8

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

Aliases: D8×C19, D4⋊C38, C81C38, C1525C2, C38.14D4, C76.17C22, (D4×C19)⋊4C2, C4.1(C2×C38), C2.3(D4×C19), SmallGroup(304,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C19
 Chief series C1 — C2 — C4 — C76 — D4×C19 — D8×C19
 Lower central C1 — C2 — C4 — D8×C19
 Upper central C1 — C38 — C76 — D8×C19

Generators and relations for D8×C19
G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D8×C19
On 152 points
Generators in S152
(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)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 32 136 101 130 93 68 52)(2 33 137 102 131 94 69 53)(3 34 138 103 132 95 70 54)(4 35 139 104 133 77 71 55)(5 36 140 105 115 78 72 56)(6 37 141 106 116 79 73 57)(7 38 142 107 117 80 74 39)(8 20 143 108 118 81 75 40)(9 21 144 109 119 82 76 41)(10 22 145 110 120 83 58 42)(11 23 146 111 121 84 59 43)(12 24 147 112 122 85 60 44)(13 25 148 113 123 86 61 45)(14 26 149 114 124 87 62 46)(15 27 150 96 125 88 63 47)(16 28 151 97 126 89 64 48)(17 29 152 98 127 90 65 49)(18 30 134 99 128 91 66 50)(19 31 135 100 129 92 67 51)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 75)(21 76)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 134)(92 135)(93 136)(94 137)(95 138)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 121)(112 122)(113 123)(114 124)

G:=sub<Sym(152)| (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,32,136,101,130,93,68,52)(2,33,137,102,131,94,69,53)(3,34,138,103,132,95,70,54)(4,35,139,104,133,77,71,55)(5,36,140,105,115,78,72,56)(6,37,141,106,116,79,73,57)(7,38,142,107,117,80,74,39)(8,20,143,108,118,81,75,40)(9,21,144,109,119,82,76,41)(10,22,145,110,120,83,58,42)(11,23,146,111,121,84,59,43)(12,24,147,112,122,85,60,44)(13,25,148,113,123,86,61,45)(14,26,149,114,124,87,62,46)(15,27,150,96,125,88,63,47)(16,28,151,97,126,89,64,48)(17,29,152,98,127,90,65,49)(18,30,134,99,128,91,66,50)(19,31,135,100,129,92,67,51), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,75)(21,76)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,134)(92,135)(93,136)(94,137)(95,138)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,121)(112,122)(113,123)(114,124)>;

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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,32,136,101,130,93,68,52)(2,33,137,102,131,94,69,53)(3,34,138,103,132,95,70,54)(4,35,139,104,133,77,71,55)(5,36,140,105,115,78,72,56)(6,37,141,106,116,79,73,57)(7,38,142,107,117,80,74,39)(8,20,143,108,118,81,75,40)(9,21,144,109,119,82,76,41)(10,22,145,110,120,83,58,42)(11,23,146,111,121,84,59,43)(12,24,147,112,122,85,60,44)(13,25,148,113,123,86,61,45)(14,26,149,114,124,87,62,46)(15,27,150,96,125,88,63,47)(16,28,151,97,126,89,64,48)(17,29,152,98,127,90,65,49)(18,30,134,99,128,91,66,50)(19,31,135,100,129,92,67,51), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,75)(21,76)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,134)(92,135)(93,136)(94,137)(95,138)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,121)(112,122)(113,123)(114,124) );

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),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,32,136,101,130,93,68,52),(2,33,137,102,131,94,69,53),(3,34,138,103,132,95,70,54),(4,35,139,104,133,77,71,55),(5,36,140,105,115,78,72,56),(6,37,141,106,116,79,73,57),(7,38,142,107,117,80,74,39),(8,20,143,108,118,81,75,40),(9,21,144,109,119,82,76,41),(10,22,145,110,120,83,58,42),(11,23,146,111,121,84,59,43),(12,24,147,112,122,85,60,44),(13,25,148,113,123,86,61,45),(14,26,149,114,124,87,62,46),(15,27,150,96,125,88,63,47),(16,28,151,97,126,89,64,48),(17,29,152,98,127,90,65,49),(18,30,134,99,128,91,66,50),(19,31,135,100,129,92,67,51)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,75),(21,76),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,134),(92,135),(93,136),(94,137),(95,138),(96,125),(97,126),(98,127),(99,128),(100,129),(101,130),(102,131),(103,132),(104,133),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,121),(112,122),(113,123),(114,124)])

133 conjugacy classes

 class 1 2A 2B 2C 4 8A 8B 19A ··· 19R 38A ··· 38R 38S ··· 38BB 76A ··· 76R 152A ··· 152AJ order 1 2 2 2 4 8 8 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 152 ··· 152 size 1 1 4 4 2 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

133 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C19 C38 C38 D4 D8 D4×C19 D8×C19 kernel D8×C19 C152 D4×C19 D8 C8 D4 C38 C19 C2 C1 # reps 1 1 2 18 18 36 1 2 18 36

Matrix representation of D8×C19 in GL2(𝔽457) generated by

 241 0 0 241
,
 377 40 377 0
,
 377 40 377 80
G:=sub<GL(2,GF(457))| [241,0,0,241],[377,377,40,0],[377,377,40,80] >;

D8×C19 in GAP, Magma, Sage, TeX

D_8\times C_{19}
% in TeX

G:=Group("D8xC19");
// GroupNames label

G:=SmallGroup(304,24);
// by ID

G=gap.SmallGroup(304,24);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-2,781,4563,2288,58]);
// Polycyclic

G:=Group<a,b,c|a^19=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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