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## G = C2×D4.10D10order 320 = 26·5

### Direct product of C2 and D4.10D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4.10D10
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C2×D4.10D10
 Lower central C5 — C10 — C2×D4.10D10
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C2×D4.10D10
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b2c, ce=ec, ede-1=d9 >

Subgroups: 2126 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C2×2- 1+4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D42D5, Q8×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C22×Dic10, C2×C4○D20, C2×D42D5, C2×Q8×D5, D4.10D10, C10×C4○D4, C2×D4.10D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C25, C22×D5, C2×2- 1+4, C23×D5, D4.10D10, D5×C24, C2×D4.10D10

Smallest permutation representation of C2×D4.10D10
On 160 points
Generators in S160
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 99)(42 100)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)
(1 131 11 121)(2 122 12 132)(3 133 13 123)(4 124 14 134)(5 135 15 125)(6 126 16 136)(7 137 17 127)(8 128 18 138)(9 139 19 129)(10 130 20 140)(21 82 31 92)(22 93 32 83)(23 84 33 94)(24 95 34 85)(25 86 35 96)(26 97 36 87)(27 88 37 98)(28 99 38 89)(29 90 39 100)(30 81 40 91)(41 146 51 156)(42 157 52 147)(43 148 53 158)(44 159 54 149)(45 150 55 160)(46 141 56 151)(47 152 57 142)(48 143 58 153)(49 154 59 144)(50 145 60 155)(61 110 71 120)(62 101 72 111)(63 112 73 102)(64 103 74 113)(65 114 75 104)(66 105 76 115)(67 116 77 106)(68 107 78 117)(69 118 79 108)(70 109 80 119)
(1 154)(2 145)(3 156)(4 147)(5 158)(6 149)(7 160)(8 151)(9 142)(10 153)(11 144)(12 155)(13 146)(14 157)(15 148)(16 159)(17 150)(18 141)(19 152)(20 143)(21 74)(22 65)(23 76)(24 67)(25 78)(26 69)(27 80)(28 71)(29 62)(30 73)(31 64)(32 75)(33 66)(34 77)(35 68)(36 79)(37 70)(38 61)(39 72)(40 63)(41 123)(42 134)(43 125)(44 136)(45 127)(46 138)(47 129)(48 140)(49 131)(50 122)(51 133)(52 124)(53 135)(54 126)(55 137)(56 128)(57 139)(58 130)(59 121)(60 132)(81 112)(82 103)(83 114)(84 105)(85 116)(86 107)(87 118)(88 109)(89 120)(90 111)(91 102)(92 113)(93 104)(94 115)(95 106)(96 117)(97 108)(98 119)(99 110)(100 101)
(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 78 11 68)(2 67 12 77)(3 76 13 66)(4 65 14 75)(5 74 15 64)(6 63 16 73)(7 72 17 62)(8 61 18 71)(9 70 19 80)(10 79 20 69)(21 148 31 158)(22 157 32 147)(23 146 33 156)(24 155 34 145)(25 144 35 154)(26 153 36 143)(27 142 37 152)(28 151 38 141)(29 160 39 150)(30 149 40 159)(41 94 51 84)(42 83 52 93)(43 92 53 82)(44 81 54 91)(45 90 55 100)(46 99 56 89)(47 88 57 98)(48 97 58 87)(49 86 59 96)(50 95 60 85)(101 127 111 137)(102 136 112 126)(103 125 113 135)(104 134 114 124)(105 123 115 133)(106 132 116 122)(107 121 117 131)(108 130 118 140)(109 139 119 129)(110 128 120 138)

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 D10 2- 1+4 D4.10D10 kernel C2×D4.10D10 C22×Dic10 C2×C4○D20 C2×D4⋊2D5 C2×Q8×D5 D4.10D10 C10×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C2 # reps 1 3 3 6 2 16 1 2 6 6 2 16 2 8

Matrix representation of C2×D4.10D10 in GL6(𝔽41)

 40 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 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 0 1 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 25 37 22 26 0 0 32 16 38 19 0 0 22 26 16 4 0 0 38 19 9 25
,
 34 7 0 0 0 0 34 1 0 0 0 0 0 0 0 29 0 36 0 0 14 14 40 40 0 0 0 36 0 12 0 0 40 40 27 27
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 35 0 0 0 25 39 7 6 0 0 35 0 39 0 0 0 7 6 16 2

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,32,22,38,0,0,37,16,26,19,0,0,22,38,16,9,0,0,26,19,4,25],[34,34,0,0,0,0,7,1,0,0,0,0,0,0,0,14,0,40,0,0,29,14,36,40,0,0,0,40,0,27,0,0,36,40,12,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,25,35,7,0,0,0,39,0,6,0,0,35,7,39,16,0,0,0,6,0,2] >;

C2×D4.10D10 in GAP, Magma, Sage, TeX

C_2\times D_4._{10}D_{10}
% in TeX

G:=Group("C2xD4.10D10");
// GroupNames label

G:=SmallGroup(320,1620);
// by ID

G=gap.SmallGroup(320,1620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,1684,102,12550]);
// Polycyclic

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

׿
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