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## G = C2×C20⋊4D4order 320 = 26·5

### Direct product of C2 and C20⋊4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20⋊4D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C20⋊4D4
 Lower central C5 — C2×C10 — C2×C20⋊4D4
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C204D4
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 2430 in 498 conjugacy classes, 159 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C42, C22×C4, C2×D4, C24, C20, D10, C2×C10, C2×C10, C2×C42, C41D4, C22×D4, D20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C41D4, C4×C20, C2×D20, C2×D20, C22×C20, C23×D5, C204D4, C2×C4×C20, C22×D20, C2×C204D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C41D4, C22×D4, D20, C22×D5, C2×C41D4, C2×D20, C23×D5, C204D4, C22×D20, C2×C204D4

Smallest permutation representation of C2×C204D4
On 160 points
Generators in S160
(1 120)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)
(1 81 39 138)(2 82 40 139)(3 83 21 140)(4 84 22 121)(5 85 23 122)(6 86 24 123)(7 87 25 124)(8 88 26 125)(9 89 27 126)(10 90 28 127)(11 91 29 128)(12 92 30 129)(13 93 31 130)(14 94 32 131)(15 95 33 132)(16 96 34 133)(17 97 35 134)(18 98 36 135)(19 99 37 136)(20 100 38 137)(41 112 145 78)(42 113 146 79)(43 114 147 80)(44 115 148 61)(45 116 149 62)(46 117 150 63)(47 118 151 64)(48 119 152 65)(49 120 153 66)(50 101 154 67)(51 102 155 68)(52 103 156 69)(53 104 157 70)(54 105 158 71)(55 106 159 72)(56 107 160 73)(57 108 141 74)(58 109 142 75)(59 110 143 76)(60 111 144 77)
(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 60)(20 59)(21 160)(22 159)(23 158)(24 157)(25 156)(26 155)(27 154)(28 153)(29 152)(30 151)(31 150)(32 149)(33 148)(34 147)(35 146)(36 145)(37 144)(38 143)(39 142)(40 141)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 100)(77 99)(78 98)(79 97)(80 96)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 2 ··· 2 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 ··· 20 2 ··· 2 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D5 D10 D10 D20 kernel C2×C20⋊4D4 C20⋊4D4 C2×C4×C20 C22×D20 C2×C20 C2×C42 C42 C22×C4 C2×C4 # reps 1 8 1 6 12 2 8 6 48

Matrix representation of C2×C204D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 2 32 0 0 0 0 37 39 0 0 0 0 0 0 2 28 0 0 0 0 13 39 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 1 0 0 0 0 5 6 0 0 0 0 0 0 0 40 0 0 0 0 1 6 0 0 0 0 0 0 25 30 0 0 0 0 27 39
,
 28 14 0 0 0 0 29 13 0 0 0 0 0 0 16 39 0 0 0 0 25 25 0 0 0 0 0 0 28 27 0 0 0 0 12 13

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,5,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,25,27,0,0,0,0,30,39],[28,29,0,0,0,0,14,13,0,0,0,0,0,0,16,25,0,0,0,0,39,25,0,0,0,0,0,0,28,12,0,0,0,0,27,13] >;

C2×C204D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_4D_4
% in TeX

G:=Group("C2xC20:4D4");
// GroupNames label

G:=SmallGroup(320,1147);
// by ID

G=gap.SmallGroup(320,1147);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,80,12550]);
// Polycyclic

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

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