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## G = C2×D20.3C4order 320 = 26·5

### Direct product of C2 and D20.3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D20.3C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×C4○D20 — C2×D20.3C4
 Lower central C5 — C10 — C2×D20.3C4
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C2×D20.3C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 718 in 266 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8○D4, C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D5×C2×C8, C2×C8⋊D5, D20.3C4, C2×C4.Dic5, C22×C40, C2×C4○D20, C2×D20.3C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D20.3C4, D5×C22×C4, C2×D20.3C4

Smallest permutation representation of C2×D20.3C4
On 160 points
Generators in S160
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 116)(22 117)(23 118)(24 119)(25 120)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 105)(22 104)(23 103)(24 102)(25 101)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 113)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)(81 154)(82 153)(83 152)(84 151)(85 150)(86 149)(87 148)(88 147)(89 146)(90 145)(91 144)(92 143)(93 142)(94 141)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)
(1 85 69 116 11 95 79 106)(2 86 70 117 12 96 80 107)(3 87 71 118 13 97 61 108)(4 88 72 119 14 98 62 109)(5 89 73 120 15 99 63 110)(6 90 74 101 16 100 64 111)(7 91 75 102 17 81 65 112)(8 92 76 103 18 82 66 113)(9 93 77 104 19 83 67 114)(10 94 78 105 20 84 68 115)(21 49 151 128 31 59 141 138)(22 50 152 129 32 60 142 139)(23 51 153 130 33 41 143 140)(24 52 154 131 34 42 144 121)(25 53 155 132 35 43 145 122)(26 54 156 133 36 44 146 123)(27 55 157 134 37 45 147 124)(28 56 158 135 38 46 148 125)(29 57 159 136 39 47 149 126)(30 58 160 137 40 48 150 127)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 8 8 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 2 10 10 10 10 2 2 1 ··· 1 2 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 D10 D10 C8○D4 C4×D5 C4×D5 D20.3C4 kernel C2×D20.3C4 D5×C2×C8 C2×C8⋊D5 D20.3C4 C2×C4.Dic5 C22×C40 C2×C4○D20 C2×Dic10 C2×D20 C4○D20 C2×C5⋊D4 C22×C8 C2×C8 C22×C4 C10 C2×C4 C23 C2 # reps 1 2 2 8 1 1 1 2 2 8 4 2 12 2 8 12 4 32

Matrix representation of C2×D20.3C4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 9 36 0 0 0 0 32 0 0 0 0 0 6 40 0 0 0 1 0
,
 1 0 0 0 0 0 9 36 0 0 0 16 32 0 0 0 0 0 6 40 0 0 0 35 35
,
 32 0 0 0 0 0 14 0 0 0 0 0 14 0 0 0 0 0 32 0 0 0 0 0 32

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,36,32,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,9,16,0,0,0,36,32,0,0,0,0,0,6,35,0,0,0,40,35],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,32,0,0,0,0,0,32] >;

C2×D20.3C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}._3C_4
% in TeX

G:=Group("C2xD20.3C4");
// GroupNames label

G:=SmallGroup(320,1410);
// by ID

G=gap.SmallGroup(320,1410);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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