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## G = C2×Dic5.14D4order 320 = 26·5

### Direct product of C2 and Dic5.14D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×Dic5.14D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×Dic5.14D4
 Lower central C5 — C2×C10 — C2×Dic5.14D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×Dic5.14D4
G = < a,b,c,d,e | a2=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede=b5d-1 >

Subgroups: 974 in 322 conjugacy classes, 135 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×30], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×12], C2×Q8 [×8], C24, Dic5 [×4], Dic5 [×6], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C2×Dic5 [×12], C2×Dic5 [×14], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×Dic5 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, Dic5.14D4 [×8], C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C22×Dic10, C23×Dic5, C2×Dic5.14D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, Dic10 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×Dic10 [×6], D4×D5 [×2], D42D5 [×2], C23×D5, Dic5.14D4 [×4], C22×Dic10, C2×D4×D5, C2×D42D5, C2×Dic5.14D4

Smallest permutation representation of C2×Dic5.14D4
On 160 points
Generators in S160
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(21 60)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 81)(38 82)(39 83)(40 84)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 141)(98 142)(99 143)(100 144)(101 137)(102 138)(103 139)(104 140)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(121 157)(122 158)(123 159)(124 160)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)
(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)
(1 103 6 108)(2 102 7 107)(3 101 8 106)(4 110 9 105)(5 109 10 104)(11 76 16 71)(12 75 17 80)(13 74 18 79)(14 73 19 78)(15 72 20 77)(21 95 26 100)(22 94 27 99)(23 93 28 98)(24 92 29 97)(25 91 30 96)(31 128 36 123)(32 127 37 122)(33 126 38 121)(34 125 39 130)(35 124 40 129)(41 115 46 120)(42 114 47 119)(43 113 48 118)(44 112 49 117)(45 111 50 116)(51 148 56 143)(52 147 57 142)(53 146 58 141)(54 145 59 150)(55 144 60 149)(61 135 66 140)(62 134 67 139)(63 133 68 138)(64 132 69 137)(65 131 70 136)(81 158 86 153)(82 157 87 152)(83 156 88 151)(84 155 89 160)(85 154 90 159)
(1 31 22 47)(2 32 23 48)(3 33 24 49)(4 34 25 50)(5 35 26 41)(6 36 27 42)(7 37 28 43)(8 38 29 44)(9 39 30 45)(10 40 21 46)(11 139 159 148)(12 140 160 149)(13 131 151 150)(14 132 152 141)(15 133 153 142)(16 134 154 143)(17 135 155 144)(18 136 156 145)(19 137 157 146)(20 138 158 147)(51 76 67 85)(52 77 68 86)(53 78 69 87)(54 79 70 88)(55 80 61 89)(56 71 62 90)(57 72 63 81)(58 73 64 82)(59 74 65 83)(60 75 66 84)(91 111 110 130)(92 112 101 121)(93 113 102 122)(94 114 103 123)(95 115 104 124)(96 116 105 125)(97 117 106 126)(98 118 107 127)(99 119 108 128)(100 120 109 129)
(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 151)(19 152)(20 153)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 41)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 D10 Dic10 D4×D5 D4⋊2D5 kernel C2×Dic5.14D4 Dic5.14D4 C2×C10.D4 C2×C4⋊Dic5 C2×C23.D5 C10×C22⋊C4 C22×Dic10 C23×Dic5 C2×Dic5 C22×C10 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C23 C22 C22 # reps 1 8 2 1 1 1 1 1 4 4 2 4 8 4 2 16 4 4

Matrix representation of C2×Dic5.14D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 34 40 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 5 6 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 2 35 0 0 0 0 21 39 0 0 0 0 0 0 20 3 0 0 0 0 3 21 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 39 9 0 0 0 0 4 2 0 0 0 0 0 0 0 40 0 0 0 0 40 0
,
 1 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 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,21,0,0,0,0,35,39,0,0,0,0,0,0,20,3,0,0,0,0,3,21,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,39,4,0,0,0,0,9,2,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C2×Dic5.14D4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5._{14}D_4
% in TeX

G:=Group("C2xDic5.14D4");
// GroupNames label

G:=SmallGroup(320,1153);
// by ID

G=gap.SmallGroup(320,1153);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,675,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=d^4=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e=b^5*d^-1>;
// generators/relations

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