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G = C2×C4×Dic5order 160 = 25·5

Direct product of C2×C4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5
 Lower central C5 — C2×C4×Dic5
 Upper central C1 — C22×C4

Generators and relations for C2×C4×Dic5
G = < a,b,c,d | a2=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 216 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×Dic5 [×12], C2×C20 [×6], C22×C10, C4×Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C42, C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C2×C4×Dic5

Smallest permutation representation of C2×C4×Dic5
Regular action on 160 points
Generators in S160
(1 68)(2 69)(3 70)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 127)(12 128)(13 129)(14 130)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 51)(30 52)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(1 43 27 34)(2 44 28 35)(3 45 29 36)(4 46 30 37)(5 47 21 38)(6 48 22 39)(7 49 23 40)(8 50 24 31)(9 41 25 32)(10 42 26 33)(11 136 156 142)(12 137 157 143)(13 138 158 144)(14 139 159 145)(15 140 160 146)(16 131 151 147)(17 132 152 148)(18 133 153 149)(19 134 154 150)(20 135 155 141)(51 71 70 87)(52 72 61 88)(53 73 62 89)(54 74 63 90)(55 75 64 81)(56 76 65 82)(57 77 66 83)(58 78 67 84)(59 79 68 85)(60 80 69 86)(91 111 107 127)(92 112 108 128)(93 113 109 129)(94 114 110 130)(95 115 101 121)(96 116 102 122)(97 117 103 123)(98 118 104 124)(99 119 105 125)(100 120 106 126)
(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)
(1 151 6 156)(2 160 7 155)(3 159 8 154)(4 158 9 153)(5 157 10 152)(11 27 16 22)(12 26 17 21)(13 25 18 30)(14 24 19 29)(15 23 20 28)(31 134 36 139)(32 133 37 138)(33 132 38 137)(34 131 39 136)(35 140 40 135)(41 149 46 144)(42 148 47 143)(43 147 48 142)(44 146 49 141)(45 145 50 150)(51 130 56 125)(52 129 57 124)(53 128 58 123)(54 127 59 122)(55 126 60 121)(61 113 66 118)(62 112 67 117)(63 111 68 116)(64 120 69 115)(65 119 70 114)(71 94 76 99)(72 93 77 98)(73 92 78 97)(74 91 79 96)(75 100 80 95)(81 106 86 101)(82 105 87 110)(83 104 88 109)(84 103 89 108)(85 102 90 107)

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

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

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

64 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4X 5A 5B 10A ··· 10N 20A ··· 20P order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 D5 Dic5 D10 D10 C4×D5 kernel C2×C4×Dic5 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 2 8 4 2 16

Matrix representation of C2×C4×Dic5 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 9 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 1 40 0 0 36 6
,
 40 0 0 0 0 40 0 0 0 0 28 32 0 0 28 13
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,28,28,0,0,32,13] >;

C2×C4×Dic5 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_5
% in TeX

G:=Group("C2xC4xDic5");
// GroupNames label

G:=SmallGroup(160,143);
// by ID

G=gap.SmallGroup(160,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,4613]);
// Polycyclic

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

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