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## G = C2×C20⋊2D4order 320 = 26·5

### Direct product of C2 and C20⋊2D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20⋊2D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×C20⋊2D4
 Lower central C5 — C2×C10 — C2×C20⋊2D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C202D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >

Subgroups: 1486 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×36], C5, C2×C4 [×6], C2×C4 [×20], D4 [×24], C23, C23 [×4], C23 [×22], D5 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24, Dic5 [×6], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×16], C2×C20 [×6], C5×D4 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4⋊D4, C4⋊Dic5 [×4], C23.D5 [×8], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5, C23×C10 [×2], C2×C4⋊Dic5, C202D4 [×8], C2×C23.D5 [×2], D5×C22×C4, C22×C5⋊D4 [×2], D4×C2×C10, C2×C202D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C4○D4 [×2], C24, D10 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊D4, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×6], C23×D5, C202D4 [×4], C2×D4×D5, C2×D42D5, C22×C5⋊D4, C2×C202D4

Smallest permutation representation of C2×C202D4
On 160 points
Generators in S160
(1 143)(2 144)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 141)(20 142)(21 98)(22 99)(23 100)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)
(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)
(1 101 29 77)(2 120 30 76)(3 119 31 75)(4 118 32 74)(5 117 33 73)(6 116 34 72)(7 115 35 71)(8 114 36 70)(9 113 37 69)(10 112 38 68)(11 111 39 67)(12 110 40 66)(13 109 21 65)(14 108 22 64)(15 107 23 63)(16 106 24 62)(17 105 25 61)(18 104 26 80)(19 103 27 79)(20 102 28 78)(41 95 121 152)(42 94 122 151)(43 93 123 150)(44 92 124 149)(45 91 125 148)(46 90 126 147)(47 89 127 146)(48 88 128 145)(49 87 129 144)(50 86 130 143)(51 85 131 142)(52 84 132 141)(53 83 133 160)(54 82 134 159)(55 81 135 158)(56 100 136 157)(57 99 137 156)(58 98 138 155)(59 97 139 154)(60 96 140 153)
(1 153)(2 142)(3 151)(4 160)(5 149)(6 158)(7 147)(8 156)(9 145)(10 154)(11 143)(12 152)(13 141)(14 150)(15 159)(16 148)(17 157)(18 146)(19 155)(20 144)(21 84)(22 93)(23 82)(24 91)(25 100)(26 89)(27 98)(28 87)(29 96)(30 85)(31 94)(32 83)(33 92)(34 81)(35 90)(36 99)(37 88)(38 97)(39 86)(40 95)(41 66)(42 75)(43 64)(44 73)(45 62)(46 71)(47 80)(48 69)(49 78)(50 67)(51 76)(52 65)(53 74)(54 63)(55 72)(56 61)(57 70)(58 79)(59 68)(60 77)(101 140)(102 129)(103 138)(104 127)(105 136)(106 125)(107 134)(108 123)(109 132)(110 121)(111 130)(112 139)(113 128)(114 137)(115 126)(116 135)(117 124)(118 133)(119 122)(120 131)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 4 4 10 10 10 10 2 2 2 2 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D4×D5 D4⋊2D5 kernel C2×C20⋊2D4 C2×C4⋊Dic5 C20⋊2D4 C2×C23.D5 D5×C22×C4 C22×C5⋊D4 D4×C2×C10 C2×C20 C22×D5 C22×D4 C2×C10 C22×C4 C2×D4 C24 C2×C4 C22 C22 # reps 1 1 8 2 1 2 1 4 4 2 4 2 8 4 16 4 4

Matrix representation of C2×C202D4 in GL5(𝔽41)

 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 40 2 0 0 0 40 1 0 0 0 0 0 7 40 0 0 0 1 0
,
 1 0 0 0 0 0 40 2 0 0 0 0 1 0 0 0 0 0 3 3 0 0 0 24 38
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 7 34 0 0 0 1 34

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],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,7,1,0,0,0,40,0],[1,0,0,0,0,0,40,0,0,0,0,2,1,0,0,0,0,0,3,24,0,0,0,3,38],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,34,34] >;

C2×C202D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_2D_4
% in TeX

G:=Group("C2xC20:2D4");
// GroupNames label

G:=SmallGroup(320,1472);
// by ID

G=gap.SmallGroup(320,1472);
# by ID

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

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𝔽