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## G = C4×D40order 320 = 26·5

### Direct product of C4 and D40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4×D40
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×D40 — C4×D40
 Lower central C5 — C10 — C20 — C4×D40
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for C4×D40
G = < a,b,c | a4=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 662 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C40, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, C4×D8, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C405C4, D205C4, C4×C40, C4×D20, C2×D40, C4×D40
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×D8, C4○D8, C4×D5, D20, C22×D5, C4×D8, D40, C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×D40, D407C2, C4×D40

Smallest permutation representation of C4×D40
On 160 points
Generators in S160
(1 101 136 55)(2 102 137 56)(3 103 138 57)(4 104 139 58)(5 105 140 59)(6 106 141 60)(7 107 142 61)(8 108 143 62)(9 109 144 63)(10 110 145 64)(11 111 146 65)(12 112 147 66)(13 113 148 67)(14 114 149 68)(15 115 150 69)(16 116 151 70)(17 117 152 71)(18 118 153 72)(19 119 154 73)(20 120 155 74)(21 81 156 75)(22 82 157 76)(23 83 158 77)(24 84 159 78)(25 85 160 79)(26 86 121 80)(27 87 122 41)(28 88 123 42)(29 89 124 43)(30 90 125 44)(31 91 126 45)(32 92 127 46)(33 93 128 47)(34 94 129 48)(35 95 130 49)(36 96 131 50)(37 97 132 51)(38 98 133 52)(39 99 134 53)(40 100 135 54)
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 2 2 2 2 4 4 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 20 20 20 20 1 1 1 1 2 2 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 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D5 D8 C4○D4 D10 D10 C4○D8 C4×D5 D20 D40 C4○D20 D40⋊7C2 kernel C4×D40 C40⋊5C4 D20⋊5C4 C4×C40 C4×D20 C2×D40 D40 C2×C20 C4×C8 C20 C20 C42 C2×C8 C10 C8 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 8 2 2 4 2 2 4 4 8 8 16 8 16

Matrix representation of C4×D40 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 40 35 0 0 6 35 0 0 0 0 18 3 0 0 38 36
,
 6 1 0 0 6 35 0 0 0 0 25 16 0 0 2 16
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[40,6,0,0,35,35,0,0,0,0,18,38,0,0,3,36],[6,6,0,0,1,35,0,0,0,0,25,2,0,0,16,16] >;

C4×D40 in GAP, Magma, Sage, TeX

C_4\times D_{40}
% in TeX

G:=Group("C4xD40");
// GroupNames label

G:=SmallGroup(320,319);
// by ID

G=gap.SmallGroup(320,319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^4=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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