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G = C2×C20⋊D4order 320 = 26·5

Direct product of C2 and C20⋊D4

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

Aliases: C2×C20⋊D4, C24.40D10, C209(C2×D4), (C2×C20)⋊13D4, (C2×D4)⋊39D10, Dic52(C2×D4), (C22×D4)⋊9D5, C102(C41D4), (C2×Dic5)⋊14D4, (D4×C10)⋊44C22, (C2×D20)⋊56C22, (C22×D20)⋊19C2, C22.149(D4×D5), (C2×C20).544C23, (C2×C10).298C24, (C4×Dic5)⋊68C22, (C22×C4).380D10, C10.145(C22×D4), (C23×C10).78C22, (C23×D5).77C22, C22.311(C23×D5), C23.135(C22×D5), (C22×C10).232C23, (C22×C20).276C22, (C2×Dic5).295C23, (C22×D5).129C23, (C22×Dic5).255C22, (D4×C2×C10)⋊6C2, C41(C2×C5⋊D4), C53(C2×C41D4), C2.105(C2×D4×D5), (C2×C4×Dic5)⋊12C2, (C2×C4)⋊10(C5⋊D4), (C2×C10).581(C2×D4), (C2×C5⋊D4)⋊47C22, (C22×C5⋊D4)⋊16C2, C2.18(C22×C5⋊D4), (C2×C4).627(C22×D5), C22.111(C2×C5⋊D4), SmallGroup(320,1475)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20⋊D4
C1C5C10C2×C10C22×D5C23×D5C22×D20 — C2×C20⋊D4
C5C2×C10 — C2×C20⋊D4
C1C23C22×D4

Generators and relations for C2×C20⋊D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=c-1 >

Subgroups: 2014 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C42, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C41D4, C22×D4, C22×D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C41D4, C4×Dic5, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×C10, C2×C4×Dic5, C20⋊D4, C22×D20, C22×C5⋊D4, D4×C2×C10, C2×C20⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C41D4, C22×D4, C5⋊D4, C22×D5, C2×C41D4, D4×D5, C2×C5⋊D4, C23×D5, C20⋊D4, C2×D4×D5, C22×C5⋊D4, C2×C20⋊D4

Smallest permutation representation of C2×C20⋊D4
On 160 points
Generators in S160
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)(81 136)(82 137)(83 138)(84 139)(85 140)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)
(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)
(1 114 123 149)(2 103 124 158)(3 112 125 147)(4 101 126 156)(5 110 127 145)(6 119 128 154)(7 108 129 143)(8 117 130 152)(9 106 131 141)(10 115 132 150)(11 104 133 159)(12 113 134 148)(13 102 135 157)(14 111 136 146)(15 120 137 155)(16 109 138 144)(17 118 139 153)(18 107 140 142)(19 116 121 151)(20 105 122 160)(21 83 79 45)(22 92 80 54)(23 81 61 43)(24 90 62 52)(25 99 63 41)(26 88 64 50)(27 97 65 59)(28 86 66 48)(29 95 67 57)(30 84 68 46)(31 93 69 55)(32 82 70 44)(33 91 71 53)(34 100 72 42)(35 89 73 51)(36 98 74 60)(37 87 75 49)(38 96 76 58)(39 85 77 47)(40 94 78 56)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4L5A5B10A···10N10O···10AD20A···20H
order12···22222222244444···45510···1010···1020···20
size11···1444420202020222210···10222···24···44···4

68 irreducible representations

dim11111122222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D10C5⋊D4D4×D5
kernelC2×C20⋊D4C2×C4×Dic5C20⋊D4C22×D20C22×C5⋊D4D4×C2×C10C2×Dic5C2×C20C22×D4C22×C4C2×D4C24C2×C4C22
# reps118141842284168

Matrix representation of C2×C20⋊D4 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
0040100
0053500
0000402
0000401
,
25320000
24160000
00211700
00152000
0000139
0000140
,
4000000
4010000
006100
0063500
000010
0000140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[25,24,0,0,0,0,32,16,0,0,0,0,0,0,21,15,0,0,0,0,17,20,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C2×C20⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes D_4
% in TeX

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

G:=SmallGroup(320,1475);
// by ID

G=gap.SmallGroup(320,1475);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,12550]);
// 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^9,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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