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## G = C2×D40⋊7C2order 320 = 26·5

### Direct product of C2 and D40⋊7C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D40⋊7C2
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×D40⋊7C2
 Lower central C5 — C10 — C20 — C2×D40⋊7C2
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C2×D407C2
G = < a,b,c,d | a2=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b20c >

Subgroups: 1054 in 266 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic5 [×4], C20 [×2], C20 [×2], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C40 [×4], Dic10 [×4], Dic10 [×2], C4×D5 [×8], D20 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×C4○D8, C40⋊C2 [×8], D40 [×4], Dic20 [×4], C2×C40 [×2], C2×C40 [×4], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20 [×2], C4○D20 [×8], C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C2×C40⋊C2 [×2], C2×D40, D407C2 [×8], C2×Dic20, C22×C40, C2×C4○D20 [×2], C2×D407C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, D20 [×4], C22×D5 [×7], C2×C4○D8, C2×D20 [×6], C23×D5, D407C2 [×2], C22×D20, C2×D407C2

Smallest permutation representation of C2×D407C2
On 160 points
Generators in S160
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 41)(37 42)(38 43)(39 44)(40 45)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 160)(120 159)
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 157)(24 158)(25 159)(26 160)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 120)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 20 20 20 20 1 1 1 1 2 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C4○D8 D20 D20 D40⋊7C2 kernel C2×D40⋊7C2 C2×C40⋊C2 C2×D40 D40⋊7C2 C2×Dic20 C22×C40 C2×C4○D20 C2×C20 C22×C10 C22×C8 C2×C8 C22×C4 C10 C2×C4 C23 C2 # reps 1 2 1 8 1 1 2 3 1 2 12 2 8 12 4 32

Matrix representation of C2×D407C2 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 8 6 0 30 38
,
 1 0 0 0 29 26 0 15 12
,
 1 0 0 0 18 1 0 5 23
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,8,30,0,6,38],[1,0,0,0,29,15,0,26,12],[1,0,0,0,18,5,0,1,23] >;

C2×D407C2 in GAP, Magma, Sage, TeX

C_2\times D_{40}\rtimes_7C_2
% in TeX

G:=Group("C2xD40:7C2");
// GroupNames label

G:=SmallGroup(320,1413);
// by ID

G=gap.SmallGroup(320,1413);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^20*c>;
// generators/relations

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