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G = C2×D20.2C4order 320 = 26·5

Direct product of C2 and D20.2C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D20.2C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×C4○D20 — C2×D20.2C4
 Lower central C5 — C10 — C2×D20.2C4
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2×D20.2C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >

Subgroups: 718 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8○D4, C8×D5, C8⋊D5, C2×C52C8, C2×C52C8, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D5×C2×C8, C2×C8⋊D5, D20.2C4, C22×C52C8, C10×M4(2), C2×C4○D20, C2×D20.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D20.2C4, D5×C22×C4, C2×D20.2C4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 2 10 10 10 10 2 2 2 ··· 2 5 ··· 5 10 10 10 10 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 D10 D10 D10 C8○D4 C4×D5 C4×D5 D20.2C4 kernel C2×D20.2C4 D5×C2×C8 C2×C8⋊D5 D20.2C4 C22×C5⋊2C8 C10×M4(2) C2×C4○D20 C2×Dic10 C2×D20 C4○D20 C2×C5⋊D4 C2×M4(2) C2×C8 M4(2) C22×C4 C10 C2×C4 C23 C2 # reps 1 2 2 8 1 1 1 2 2 8 4 2 4 8 2 8 12 4 8

Matrix representation of C2×D20.2C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 34 40 0 0 1 0 0 0 0 0 32 0 0 0 23 9
,
 7 1 0 0 34 34 0 0 0 0 40 1 0 0 0 1
,
 32 0 0 0 0 32 0 0 0 0 14 27 0 0 28 27
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,40,0,0,0,0,0,32,23,0,0,0,9],[7,34,0,0,1,34,0,0,0,0,40,0,0,0,1,1],[32,0,0,0,0,32,0,0,0,0,14,28,0,0,27,27] >;

C2×D20.2C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}._2C_4
% in TeX

G:=Group("C2xD20.2C4");
// GroupNames label

G:=SmallGroup(320,1416);
// by ID

G=gap.SmallGroup(320,1416);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,297,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations

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