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## G = (C2×C20).290D4order 320 = 26·5

### 264th non-split extension by C2×C20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C20).290D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C2×C20).290D4
 Lower central C5 — C22×C10 — (C2×C20).290D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C20).290D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab9, dcd=ac-1 >

Subgroups: 726 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.11D4, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C10.10C42, C2×D10⋊C4, C2×D10⋊C4, C10×C4⋊C4, (C2×C20).290D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C422C2, C5⋊D4, C22×D5, C23.11D4, C4○D20, D4×D5, D42D5, Q82D5, C2×C5⋊D4, D10.13D4, C4⋊C4⋊D5, C23.23D10, C202D4, C20.23D4, (C2×C20).290D4

Smallest permutation representation of (C2×C20).290D4
On 160 points
Generators in S160
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(61 137)(62 138)(63 139)(64 140)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(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 85 153 76)(2 37 154 131)(3 83 155 74)(4 35 156 129)(5 81 157 72)(6 33 158 127)(7 99 159 70)(8 31 160 125)(9 97 141 68)(10 29 142 123)(11 95 143 66)(12 27 144 121)(13 93 145 64)(14 25 146 139)(15 91 147 62)(16 23 148 137)(17 89 149 80)(18 21 150 135)(19 87 151 78)(20 39 152 133)(22 119 136 45)(24 117 138 43)(26 115 140 41)(28 113 122 59)(30 111 124 57)(32 109 126 55)(34 107 128 53)(36 105 130 51)(38 103 132 49)(40 101 134 47)(42 92 116 63)(44 90 118 61)(46 88 120 79)(48 86 102 77)(50 84 104 75)(52 82 106 73)(54 100 108 71)(56 98 110 69)(58 96 112 67)(60 94 114 65)
(2 58)(3 19)(4 56)(5 17)(6 54)(7 15)(8 52)(9 13)(10 50)(12 48)(14 46)(16 44)(18 42)(20 60)(21 139)(22 72)(23 137)(24 70)(25 135)(26 68)(27 133)(28 66)(29 131)(30 64)(31 129)(32 62)(33 127)(34 80)(35 125)(36 78)(37 123)(38 76)(39 121)(40 74)(41 57)(43 55)(45 53)(47 51)(61 90)(63 88)(65 86)(67 84)(69 82)(71 100)(73 98)(75 96)(77 94)(79 92)(81 136)(83 134)(85 132)(87 130)(89 128)(91 126)(93 124)(95 122)(97 140)(99 138)(101 105)(102 144)(104 142)(106 160)(107 119)(108 158)(109 117)(110 156)(111 115)(112 154)(114 152)(116 150)(118 148)(120 146)(141 145)(147 159)(149 157)(151 155)

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4F 4G ··· 4L 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 D4 D4 D5 C4○D4 D10 C5⋊D4 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel (C2×C20).290D4 C10.10C42 C2×D10⋊C4 C10×C4⋊C4 C2×C20 C22×D5 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 3 1 2 2 2 10 6 8 16 2 2 4

Matrix representation of (C2×C20).290D4 in GL6(𝔽41)

 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
,
 14 22 0 0 0 0 19 27 0 0 0 0 0 0 35 23 0 0 0 0 18 20 0 0 0 0 0 0 40 40 0 0 0 0 36 35
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 13 28 0 0 0 0 32 28 0 0 0 0 0 0 20 3 0 0 0 0 3 21
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 35 0 0 0 0 40 35 0 0 0 0 0 0 6 7 0 0 0 0 36 35

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],[14,19,0,0,0,0,22,27,0,0,0,0,0,0,35,18,0,0,0,0,23,20,0,0,0,0,0,0,40,36,0,0,0,0,40,35],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,32,0,0,0,0,28,28,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,6,36,0,0,0,0,7,35] >;

(C2×C20).290D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{290}D_4
% in TeX

G:=Group("(C2xC20).290D4");
// GroupNames label

G:=SmallGroup(320,620);
// by ID

G=gap.SmallGroup(320,620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,100,12550]);
// 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=a*b^-1,d*b*d=a*b^9,d*c*d=a*c^-1>;
// generators/relations

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