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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 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], D5 [×4], C10, C10 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], Dic5 [×4], C20 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×4], C22×D5 [×12], C22×C10, C2×C4.4D4, C4×Dic5 [×4], D10⋊C4 [×16], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23×D5 [×2], C2×C4×Dic5, C2×D10⋊C4 [×4], C20.23D4 [×8], C22×D20, Q8×C2×C10, C2×C20.23D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C4.4D4, Q82D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20.23D4 [×4], C2×Q82D5 [×2], C22×C5⋊D4, C2×C20.23D4

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

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

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

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

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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