Copied to
clipboard

## G = C22×D40order 320 = 26·5

### Direct product of C22 and D40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C22×D40
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C22×D20 — C22×D40
 Lower central C5 — C10 — C20 — C22×D40
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C22×D40
G = < a,b,c,d | a2=b2=c40=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1822 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C2×C8, D8, C22×C4, C2×D4, C24, C20, C20, D10, C2×C10, C22×C8, C2×D8, C22×D4, C40, D20, D20, C2×C20, C22×D5, C22×C10, C22×D8, D40, C2×C40, C2×D20, C2×D20, C22×C20, C23×D5, C2×D40, C22×C40, C22×D20, C22×D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C24, D10, C2×D8, C22×D4, D20, C22×D5, C22×D8, D40, C2×D20, C23×D5, C2×D40, C22×D20, C22×D40

Smallest permutation representation of C22×D40
On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 81)(37 82)(38 83)(39 84)(40 85)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 121)(80 122)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 41)(38 42)(39 43)(40 44)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 121)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(81 156)(82 155)(83 154)(84 153)(85 152)(86 151)(87 150)(88 149)(89 148)(90 147)(91 146)(92 145)(93 144)(94 143)(95 142)(96 141)(97 140)(98 139)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 160)(118 159)(119 158)(120 157)

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

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

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

92 conjugacy classes

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

92 irreducible representations

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

Matrix representation of C22×D40 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 39 3 0 0 17 15
,
 1 0 0 0 0 40 0 0 0 0 7 1 0 0 34 34
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,39,17,0,0,3,15],[1,0,0,0,0,40,0,0,0,0,7,34,0,0,1,34] >;

C22×D40 in GAP, Magma, Sage, TeX

C_2^2\times D_{40}
% in TeX

G:=Group("C2^2xD40");
// GroupNames label

G:=SmallGroup(320,1412);
// by ID

G=gap.SmallGroup(320,1412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,192,1684,102,12550]);
// Polycyclic

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

׿
×
𝔽