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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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×6], D4 [×6], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×8], C2×C10, C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C40 [×2], C40, C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4×D8, D40 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20 [×2], C405C4, D205C4 [×2], C4×C40, C4×D20 [×2], C2×D40, C4×D40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], D20 [×2], C22×D5, C4×D8, D40 [×2], 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 58 97 155)(2 59 98 156)(3 60 99 157)(4 61 100 158)(5 62 101 159)(6 63 102 160)(7 64 103 121)(8 65 104 122)(9 66 105 123)(10 67 106 124)(11 68 107 125)(12 69 108 126)(13 70 109 127)(14 71 110 128)(15 72 111 129)(16 73 112 130)(17 74 113 131)(18 75 114 132)(19 76 115 133)(20 77 116 134)(21 78 117 135)(22 79 118 136)(23 80 119 137)(24 41 120 138)(25 42 81 139)(26 43 82 140)(27 44 83 141)(28 45 84 142)(29 46 85 143)(30 47 86 144)(31 48 87 145)(32 49 88 146)(33 50 89 147)(34 51 90 148)(35 52 91 149)(36 53 92 150)(37 54 93 151)(38 55 94 152)(39 56 95 153)(40 57 96 154)
(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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 99)(14 98)(15 97)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 87)(26 86)(27 85)(28 84)(29 83)(30 82)(31 81)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 128)(60 127)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)

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

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

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

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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