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## G = C2×C20.23D4order 320 = 26·5

### Direct product of C2 and C20.23D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20.23D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C20.23D4
 Lower central C5 — C2×C10 — C2×C20.23D4
 Upper central C1 — C23 — C22×Q8

Generators and relations for C2×C20.23D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=b10c-1 >

Subgroups: 1278 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, C2×Q8, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C2×C4.4D4, C4×Dic5, D10⋊C4, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, Q8×C10, Q8×C10, C23×D5, C2×C4×Dic5, C2×D10⋊C4, C20.23D4, C22×D20, Q8×C2×C10, C2×C20.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4.4D4, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C2×C4.4D4, Q82D5, C2×C5⋊D4, C23×D5, C20.23D4, C2×Q82D5, C22×C5⋊D4, C2×C20.23D4

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 20 20 2 2 2 2 4 4 4 4 10 ··· 10 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C5⋊D4 Q8⋊2D5 kernel C2×C20.23D4 C2×C4×Dic5 C2×D10⋊C4 C20.23D4 C22×D20 Q8×C2×C10 C2×C20 C22×Q8 C2×C10 C22×C4 C2×Q8 C2×C4 C22 # reps 1 1 4 8 1 1 4 2 8 6 8 16 8

Matrix representation of C2×C20.23D4 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 40 0 0 0 0 0 0 40
,
 34 33 0 0 0 0 1 1 0 0 0 0 0 0 6 40 0 0 0 0 36 1 0 0 0 0 0 0 0 32 0 0 0 0 32 0
,
 38 37 0 0 0 0 23 3 0 0 0 0 0 0 20 3 0 0 0 0 3 21 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 7 7 0 0 0 0 40 34 0 0 0 0 0 0 35 34 0 0 0 0 5 6 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,33,1,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[38,23,0,0,0,0,37,3,0,0,0,0,0,0,20,3,0,0,0,0,3,21,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C2×C20.23D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{23}D_4
% in TeX

G:=Group("C2xC20.23D4");
// GroupNames label

G:=SmallGroup(320,1486);
// by ID

G=gap.SmallGroup(320,1486);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,136,12550]);
// 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^9,d*b*d=b^-1,d*c*d=b^10*c^-1>;
// generators/relations

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