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G = C2×C4.D20order 320 = 26·5

Direct product of C2 and C4.D20

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

Aliases: C2×C4.D20, C4240D10, C4.43(C2×D20), (C2×C4).98D20, (C2×C42)⋊10D5, (C4×C20)⋊51C22, (C2×C20).389D4, C20.286(C2×D4), C10.4(C22×D4), C2.6(C22×D20), C101(C4.4D4), (C2×C10).20C24, (C22×D20).8C2, C22.65(C2×D20), (C2×C20).781C23, (C22×Dic10)⋊4C2, (C22×C4).439D10, (C2×Dic5).4C23, D10⋊C439C22, (C22×D5).2C23, C22.63(C23×D5), (C2×Dic10)⋊46C22, (C2×D20).210C22, C22.69(C4○D20), (C23×D5).28C22, C23.316(C22×D5), (C22×C20).504C22, (C22×C10).382C23, (C22×Dic5).74C22, (C2×C4×C20)⋊9C2, C51(C2×C4.4D4), C10.7(C2×C4○D4), C2.9(C2×C4○D20), (C2×C10).171(C2×D4), (C2×D10⋊C4)⋊12C2, (C2×C10).97(C4○D4), (C2×C4).649(C22×D5), SmallGroup(320,1148)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C4.D20
C1C5C10C2×C10C22×D5C23×D5C2×D10⋊C4 — C2×C4.D20
C5C2×C10 — C2×C4.D20
C1C23C2×C42

Generators and relations for C2×C4.D20
 G = < a,b,c,d | a2=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 1374 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4.4D4, D10⋊C4, C4×C20, C2×Dic10, C2×Dic10, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C4.D20, C2×D10⋊C4, C2×C4×C20, C22×Dic10, C22×D20, C2×C4.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4.4D4, C22×D4, C2×C4○D4, D20, C22×D5, C2×C4.4D4, C2×D20, C4○D20, C23×D5, C4.D20, C22×D20, C2×C4○D20, C2×C4.D20

Smallest permutation representation of C2×C4.D20
On 160 points
Generators in S160
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(41 159)(42 160)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 121)(80 122)(81 119)(82 120)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)
(1 100 65 148)(2 81 66 149)(3 82 67 150)(4 83 68 151)(5 84 69 152)(6 85 70 153)(7 86 71 154)(8 87 72 155)(9 88 73 156)(10 89 74 157)(11 90 75 158)(12 91 76 159)(13 92 77 160)(14 93 78 141)(15 94 79 142)(16 95 80 143)(17 96 61 144)(18 97 62 145)(19 98 63 146)(20 99 64 147)(21 105 134 57)(22 106 135 58)(23 107 136 59)(24 108 137 60)(25 109 138 41)(26 110 139 42)(27 111 140 43)(28 112 121 44)(29 113 122 45)(30 114 123 46)(31 115 124 47)(32 116 125 48)(33 117 126 49)(34 118 127 50)(35 119 128 51)(36 120 129 52)(37 101 130 53)(38 102 131 54)(39 103 132 55)(40 104 133 56)
(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 49 65 117)(2 116 66 48)(3 47 67 115)(4 114 68 46)(5 45 69 113)(6 112 70 44)(7 43 71 111)(8 110 72 42)(9 41 73 109)(10 108 74 60)(11 59 75 107)(12 106 76 58)(13 57 77 105)(14 104 78 56)(15 55 79 103)(16 102 80 54)(17 53 61 101)(18 120 62 52)(19 51 63 119)(20 118 64 50)(21 92 134 160)(22 159 135 91)(23 90 136 158)(24 157 137 89)(25 88 138 156)(26 155 139 87)(27 86 140 154)(28 153 121 85)(29 84 122 152)(30 151 123 83)(31 82 124 150)(32 149 125 81)(33 100 126 148)(34 147 127 99)(35 98 128 146)(36 145 129 97)(37 96 130 144)(38 143 131 95)(39 94 132 142)(40 141 133 93)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M4N4O4P5A5B10A···10N20A···20AV
order12···222224···444445510···1020···20
size11···1202020202···220202020222···22···2

92 irreducible representations

dim1111112222222
type+++++++++++
imageC1C2C2C2C2C2D4D5C4○D4D10D10D20C4○D20
kernelC2×C4.D20C4.D20C2×D10⋊C4C2×C4×C20C22×Dic10C22×D20C2×C20C2×C42C2×C10C42C22×C4C2×C4C22
# reps184111428861632

Matrix representation of C2×C4.D20 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
0303200
091100
0001119
0001330
,
10000
022900
032000
000321
0002324
,
400000
0381700
038300
000022
000130

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],[40,0,0,0,0,0,30,9,0,0,0,32,11,0,0,0,0,0,11,13,0,0,0,19,30],[1,0,0,0,0,0,22,32,0,0,0,9,0,0,0,0,0,0,3,23,0,0,0,21,24],[40,0,0,0,0,0,38,38,0,0,0,17,3,0,0,0,0,0,0,13,0,0,0,22,0] >;

C2×C4.D20 in GAP, Magma, Sage, TeX

C_2\times C_4.D_{20}
% in TeX

G:=Group("C2xC4.D20");
// GroupNames label

G:=SmallGroup(320,1148);
// by ID

G=gap.SmallGroup(320,1148);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,136,12550]);
// Polycyclic

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

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