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## G = C2×D10⋊C8order 320 = 26·5

### Direct product of C2 and D10⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D10⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C2×D10⋊C8
 Lower central C5 — C10 — C2×D10⋊C8
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D10⋊C8
G = < a,b,c,d | a2=b10=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b7c >

Subgroups: 762 in 202 conjugacy classes, 76 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×6], C22, C22 [×6], C22 [×16], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C2×C8 [×8], C22×C4, C22×C4 [×9], C24, Dic5 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C22⋊C8 [×4], C22×C8 [×2], C23×C4, C5⋊C8 [×4], C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊C8, C2×C5⋊C8 [×4], C2×C5⋊C8 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊C8 [×4], C22×C5⋊C8 [×2], D5×C22×C4, C2×D10⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], F5, C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×F5 [×3], C2×C22⋊C8, D5⋊C8 [×2], C4.F5 [×2], C22⋊F5 [×2], C22×F5, D10⋊C8 [×4], C2×D5⋊C8, C2×C4.F5, C2×C22⋊F5, C2×D10⋊C8

Smallest permutation representation of C2×D10⋊C8
On 160 points
Generators in S160
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 61)(9 62)(10 63)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(21 57)(22 58)(23 59)(24 60)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 90)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(91 147)(92 148)(93 149)(94 150)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 131)(108 132)(109 133)(110 134)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 151)
(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)(151 152 153 154 155 156 157 158 159 160)
(1 54)(2 53)(3 52)(4 51)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 130)(18 129)(19 128)(20 127)(21 61)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 73)(32 72)(33 71)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 83)(42 82)(43 81)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 140)(98 139)(99 138)(100 137)(101 148)(102 147)(103 146)(104 145)(105 144)(106 143)(107 142)(108 141)(109 150)(110 149)(111 160)(112 159)(113 158)(114 157)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)
(1 99 35 116 24 105 46 125)(2 96 34 119 25 102 45 128)(3 93 33 112 26 109 44 121)(4 100 32 115 27 106 43 124)(5 97 31 118 28 103 42 127)(6 94 40 111 29 110 41 130)(7 91 39 114 30 107 50 123)(8 98 38 117 21 104 49 126)(9 95 37 120 22 101 48 129)(10 92 36 113 23 108 47 122)(11 56 131 79 154 70 147 88)(12 53 140 72 155 67 146 81)(13 60 139 75 156 64 145 84)(14 57 138 78 157 61 144 87)(15 54 137 71 158 68 143 90)(16 51 136 74 159 65 142 83)(17 58 135 77 160 62 141 86)(18 55 134 80 151 69 150 89)(19 52 133 73 152 66 149 82)(20 59 132 76 153 63 148 85)

G:=sub<Sym(160)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,61)(9,62)(10,63)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,57)(22,58)(23,59)(24,60)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,90)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(91,147)(92,148)(93,149)(94,150)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,131)(108,132)(109,133)(110,134)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,151), (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)(151,152,153,154,155,156,157,158,159,160), (1,54)(2,53)(3,52)(4,51)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,130)(18,129)(19,128)(20,127)(21,61)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,73)(32,72)(33,71)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,83)(42,82)(43,81)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,140)(98,139)(99,138)(100,137)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,150)(110,149)(111,160)(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)(119,152)(120,151), (1,99,35,116,24,105,46,125)(2,96,34,119,25,102,45,128)(3,93,33,112,26,109,44,121)(4,100,32,115,27,106,43,124)(5,97,31,118,28,103,42,127)(6,94,40,111,29,110,41,130)(7,91,39,114,30,107,50,123)(8,98,38,117,21,104,49,126)(9,95,37,120,22,101,48,129)(10,92,36,113,23,108,47,122)(11,56,131,79,154,70,147,88)(12,53,140,72,155,67,146,81)(13,60,139,75,156,64,145,84)(14,57,138,78,157,61,144,87)(15,54,137,71,158,68,143,90)(16,51,136,74,159,65,142,83)(17,58,135,77,160,62,141,86)(18,55,134,80,151,69,150,89)(19,52,133,73,152,66,149,82)(20,59,132,76,153,63,148,85)>;

G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,61)(9,62)(10,63)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,57)(22,58)(23,59)(24,60)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,90)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(91,147)(92,148)(93,149)(94,150)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,131)(108,132)(109,133)(110,134)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,151), (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)(151,152,153,154,155,156,157,158,159,160), (1,54)(2,53)(3,52)(4,51)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,130)(18,129)(19,128)(20,127)(21,61)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,73)(32,72)(33,71)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,83)(42,82)(43,81)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,140)(98,139)(99,138)(100,137)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,150)(110,149)(111,160)(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)(119,152)(120,151), (1,99,35,116,24,105,46,125)(2,96,34,119,25,102,45,128)(3,93,33,112,26,109,44,121)(4,100,32,115,27,106,43,124)(5,97,31,118,28,103,42,127)(6,94,40,111,29,110,41,130)(7,91,39,114,30,107,50,123)(8,98,38,117,21,104,49,126)(9,95,37,120,22,101,48,129)(10,92,36,113,23,108,47,122)(11,56,131,79,154,70,147,88)(12,53,140,72,155,67,146,81)(13,60,139,75,156,64,145,84)(14,57,138,78,157,61,144,87)(15,54,137,71,158,68,143,90)(16,51,136,74,159,65,142,83)(17,58,135,77,160,62,141,86)(18,55,134,80,151,69,150,89)(19,52,133,73,152,66,149,82)(20,59,132,76,153,63,148,85) );

G=PermutationGroup([(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,61),(9,62),(10,63),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,111),(19,112),(20,113),(21,57),(22,58),(23,59),(24,60),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,90),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,80),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(91,147),(92,148),(93,149),(94,150),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,131),(108,132),(109,133),(110,134),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,151)], [(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),(151,152,153,154,155,156,157,158,159,160)], [(1,54),(2,53),(3,52),(4,51),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,126),(12,125),(13,124),(14,123),(15,122),(16,121),(17,130),(18,129),(19,128),(20,127),(21,61),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,73),(32,72),(33,71),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,83),(42,82),(43,81),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,84),(91,136),(92,135),(93,134),(94,133),(95,132),(96,131),(97,140),(98,139),(99,138),(100,137),(101,148),(102,147),(103,146),(104,145),(105,144),(106,143),(107,142),(108,141),(109,150),(110,149),(111,160),(112,159),(113,158),(114,157),(115,156),(116,155),(117,154),(118,153),(119,152),(120,151)], [(1,99,35,116,24,105,46,125),(2,96,34,119,25,102,45,128),(3,93,33,112,26,109,44,121),(4,100,32,115,27,106,43,124),(5,97,31,118,28,103,42,127),(6,94,40,111,29,110,41,130),(7,91,39,114,30,107,50,123),(8,98,38,117,21,104,49,126),(9,95,37,120,22,101,48,129),(10,92,36,113,23,108,47,122),(11,56,131,79,154,70,147,88),(12,53,140,72,155,67,146,81),(13,60,139,75,156,64,145,84),(14,57,138,78,157,61,144,87),(15,54,137,71,158,68,143,90),(16,51,136,74,159,65,142,83),(17,58,135,77,160,62,141,86),(18,55,134,80,151,69,150,89),(19,52,133,73,152,66,149,82),(20,59,132,76,153,63,148,85)])

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 5 8A ··· 8P 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 2 2 2 2 5 ··· 5 4 10 ··· 10 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 D4 M4(2) F5 C2×F5 C2×F5 D5⋊C8 C4.F5 C22⋊F5 kernel C2×D10⋊C8 D10⋊C8 C22×C5⋊C8 D5×C22×C4 C2×C4×D5 C22×C20 C23×D5 C22×D5 C2×Dic5 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 4 2 2 16 4 4 1 2 1 4 4 4

Matrix representation of C2×D10⋊C8 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 40 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0
,
 1 0 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 0 0 0 0 1 40 0 0 0 0 0 1 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 0 0 40
,
 1 39 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 1 39 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 32 25 32 17 0 0 0 0 23 1 8 8 0 0 0 0 40 33 33 40 0 0 0 0 24 24 9 16

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[1,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40],[1,1,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,32,23,40,24,0,0,0,0,25,1,33,24,0,0,0,0,32,8,33,9,0,0,0,0,17,8,40,16] >;

C2×D10⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes C_8
% in TeX

G:=Group("C2xD10:C8");
// GroupNames label

G:=SmallGroup(320,1089);
// by ID

G=gap.SmallGroup(320,1089);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^7*c>;
// generators/relations

׿
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