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## G = C2×D4.Dic5order 320 = 26·5

### Direct product of C2 and D4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×D4.Dic5
 Lower central C5 — C10 — C2×D4.Dic5
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C2×D4.Dic5
G = < a,b,c,d,e | a2=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Subgroups: 494 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C52C8 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C2×C8○D4, C2×C52C8, C2×C52C8 [×15], C4.Dic5 [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×C52C8 [×3], C2×C4.Dic5 [×3], D4.Dic5 [×8], C10×C4○D4, C2×D4.Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C8○D4 [×2], C23×C4, C2×Dic5 [×28], C22×D5 [×7], C2×C8○D4, C22×Dic5 [×14], C23×D5, D4.Dic5 [×2], C23×Dic5, C2×D4.Dic5

Smallest permutation representation of C2×D4.Dic5
On 160 points
Generators in S160
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 118)(82 119)(83 120)(84 101)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)(121 159)(122 160)(123 141)(124 142)(125 143)(126 144)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 76 31 66)(22 77 32 67)(23 78 33 68)(24 79 34 69)(25 80 35 70)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)(81 140 91 130)(82 121 92 131)(83 122 93 132)(84 123 94 133)(85 124 95 134)(86 125 96 135)(87 126 97 136)(88 127 98 137)(89 128 99 138)(90 129 100 139)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)(106 146 116 156)(107 147 117 157)(108 148 118 158)(109 149 119 159)(110 150 120 160)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 136 131 126)(122 137 132 127)(123 138 133 128)(124 139 134 129)(125 140 135 130)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)
(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)
(1 118 6 103 11 108 16 113)(2 107 7 112 12 117 17 102)(3 116 8 101 13 106 18 111)(4 105 9 110 14 115 19 120)(5 114 10 119 15 104 20 109)(21 86 26 91 31 96 36 81)(22 95 27 100 32 85 37 90)(23 84 28 89 33 94 38 99)(24 93 29 98 34 83 39 88)(25 82 30 87 35 92 40 97)(41 148 46 153 51 158 56 143)(42 157 47 142 52 147 57 152)(43 146 48 151 53 156 58 141)(44 155 49 160 54 145 59 150)(45 144 50 149 55 154 60 159)(61 130 66 135 71 140 76 125)(62 139 67 124 72 129 77 134)(63 128 68 133 73 138 78 123)(64 137 69 122 74 127 79 132)(65 126 70 131 75 136 80 121)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 5A 5B 8A ··· 8H 8I ··· 8T 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 2 5 ··· 5 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D5 D10 Dic5 Dic5 Dic5 D10 C8○D4 D4.Dic5 kernel C2×D4.Dic5 C22×C5⋊2C8 C2×C4.Dic5 D4.Dic5 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C10 C2 # reps 1 3 3 8 1 6 2 8 2 6 6 2 8 8 8 8

Matrix representation of C2×D4.Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 0 9 0 0 9 0
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 32
,
 40 40 0 0 8 7 0 0 0 0 9 0 0 0 0 9
,
 9 25 0 0 5 32 0 0 0 0 3 0 0 0 0 3
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[40,8,0,0,40,7,0,0,0,0,9,0,0,0,0,9],[9,5,0,0,25,32,0,0,0,0,3,0,0,0,0,3] >;

C2×D4.Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4.{\rm Dic}_5
% in TeX

G:=Group("C2xD4.Dic5");
// GroupNames label

G:=SmallGroup(320,1490);
// by ID

G=gap.SmallGroup(320,1490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^10=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations

׿
×
𝔽