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G = C2×C202D4order 320 = 26·5

Direct product of C2 and C202D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C202D4, C24.39D10, C207(C2×D4), D104(C2×D4), (C2×C20)⋊12D4, (C2×D4)⋊36D10, (C22×D4)⋊6D5, C104(C4⋊D4), (C22×D5)⋊12D4, (D4×C10)⋊43C22, C4⋊Dic577C22, C22.147(D4×D5), (C2×C20).542C23, (C2×C10).295C24, (C22×C4).379D10, C10.142(C22×D4), C23.D561C22, (C23×C10).76C22, C23.337(C22×D5), C22.308(C23×D5), C22.79(D42D5), (C22×C20).275C22, (C22×C10).419C23, (C2×Dic5).152C23, (C22×D5).249C23, (C23×D5).125C22, (C22×Dic5).163C22, (D4×C2×C10)⋊4C2, C55(C2×C4⋊D4), C43(C2×C5⋊D4), C2.102(C2×D4×D5), (D5×C22×C4)⋊6C2, (C2×C4×D5)⋊57C22, (C2×C4)⋊13(C5⋊D4), (C2×C4⋊Dic5)⋊45C2, C10.105(C2×C4○D4), C2.69(C2×D42D5), (C2×C10).580(C2×D4), (C2×C5⋊D4)⋊44C22, (C22×C5⋊D4)⋊13C2, (C2×C23.D5)⋊28C2, C2.15(C22×C5⋊D4), (C2×C4).625(C22×D5), C22.110(C2×C5⋊D4), (C2×C10).177(C4○D4), SmallGroup(320,1472)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C202D4
C1C5C10C2×C10C22×D5C23×D5D5×C22×C4 — C2×C202D4
C5C2×C10 — C2×C202D4
C1C23C22×D4

Generators and relations for C2×C202D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >

Subgroups: 1486 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4⋊D4, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×C10, C2×C4⋊Dic5, C202D4, C2×C23.D5, D5×C22×C4, C22×C5⋊D4, D4×C2×C10, C2×C202D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C4⋊D4, D4×D5, D42D5, C2×C5⋊D4, C23×D5, C202D4, C2×D4×D5, C2×D42D5, C22×C5⋊D4, C2×C202D4

Smallest permutation representation of C2×C202D4
On 160 points
Generators in S160
(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 41)(18 42)(19 43)(20 44)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)
(1 101 83 133)(2 120 84 132)(3 119 85 131)(4 118 86 130)(5 117 87 129)(6 116 88 128)(7 115 89 127)(8 114 90 126)(9 113 91 125)(10 112 92 124)(11 111 93 123)(12 110 94 122)(13 109 95 121)(14 108 96 140)(15 107 97 139)(16 106 98 138)(17 105 99 137)(18 104 100 136)(19 103 81 135)(20 102 82 134)(21 45 141 77)(22 44 142 76)(23 43 143 75)(24 42 144 74)(25 41 145 73)(26 60 146 72)(27 59 147 71)(28 58 148 70)(29 57 149 69)(30 56 150 68)(31 55 151 67)(32 54 152 66)(33 53 153 65)(34 52 154 64)(35 51 155 63)(36 50 156 62)(37 49 157 61)(38 48 158 80)(39 47 159 79)(40 46 160 78)
(1 55)(2 44)(3 53)(4 42)(5 51)(6 60)(7 49)(8 58)(9 47)(10 56)(11 45)(12 54)(13 43)(14 52)(15 41)(16 50)(17 59)(18 48)(19 57)(20 46)(21 111)(22 120)(23 109)(24 118)(25 107)(26 116)(27 105)(28 114)(29 103)(30 112)(31 101)(32 110)(33 119)(34 108)(35 117)(36 106)(37 115)(38 104)(39 113)(40 102)(61 89)(62 98)(63 87)(64 96)(65 85)(66 94)(67 83)(68 92)(69 81)(70 90)(71 99)(72 88)(73 97)(74 86)(75 95)(76 84)(77 93)(78 82)(79 91)(80 100)(121 143)(122 152)(123 141)(124 150)(125 159)(126 148)(127 157)(128 146)(129 155)(130 144)(131 153)(132 142)(133 151)(134 160)(135 149)(136 158)(137 147)(138 156)(139 145)(140 154)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10N10O···10AD20A···20H
order12···2222222224444444444445510···1010···1020···20
size11···144441010101022221010101020202020222···24···44···4

68 irreducible representations

dim11111112222222244
type++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C5⋊D4D4×D5D42D5
kernelC2×C202D4C2×C4⋊Dic5C202D4C2×C23.D5D5×C22×C4C22×C5⋊D4D4×C2×C10C2×C20C22×D5C22×D4C2×C10C22×C4C2×D4C24C2×C4C22C22
# reps118212144242841644

Matrix representation of C2×C202D4 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
040200
040100
000740
00010
,
10000
040200
00100
00033
0002438
,
10000
040000
004000
000734
000134

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,7,1,0,0,0,40,0],[1,0,0,0,0,0,40,0,0,0,0,2,1,0,0,0,0,0,3,24,0,0,0,3,38],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,34,34] >;

C2×C202D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_2D_4
% in TeX

G:=Group("C2xC20:2D4");
// GroupNames label

G:=SmallGroup(320,1472);
// by ID

G=gap.SmallGroup(320,1472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

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