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## G = C2×Q8.D10order 320 = 26·5

### Direct product of C2 and Q8.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×Q8.D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×Q8⋊2D5 — C2×Q8.D10
 Lower central C5 — C10 — C20 — C2×Q8.D10
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×Q8.D10
G = < a,b,c,d,e | a2=b4=e2=1, c2=d10=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d9 >

Subgroups: 1054 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C52C8, C40, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, C2×C4○D8, C8×D5, D40, C2×C52C8, Q8⋊D5, C2×C40, C5×Q16, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5, Q82D5, Q8×C10, D5×C2×C8, C2×D40, Q8.D10, C2×Q8⋊D5, C10×Q16, C2×Q82D5, C2×Q8.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C4○D8, C22×D4, C22×D5, C2×C4○D8, D4×D5, C23×D5, Q8.D10, C2×D4×D5, C2×Q8.D10

Smallest permutation representation of C2×Q8.D10
On 160 points
Generators in S160
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 121)(38 122)(39 123)(40 124)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 156)(82 157)(83 158)(84 159)(85 160)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)
(1 30 11 40)(2 21 12 31)(3 32 13 22)(4 23 14 33)(5 34 15 24)(6 25 16 35)(7 36 17 26)(8 27 18 37)(9 38 19 28)(10 29 20 39)(41 130 51 140)(42 121 52 131)(43 132 53 122)(44 123 54 133)(45 134 55 124)(46 125 56 135)(47 136 57 126)(48 127 58 137)(49 138 59 128)(50 129 60 139)(61 145 71 155)(62 156 72 146)(63 147 73 157)(64 158 74 148)(65 149 75 159)(66 160 76 150)(67 151 77 141)(68 142 78 152)(69 153 79 143)(70 144 80 154)(81 102 91 112)(82 113 92 103)(83 104 93 114)(84 115 94 105)(85 106 95 116)(86 117 96 107)(87 108 97 118)(88 119 98 109)(89 110 99 120)(90 101 100 111)
(1 142 11 152)(2 79 12 69)(3 144 13 154)(4 61 14 71)(5 146 15 156)(6 63 16 73)(7 148 17 158)(8 65 18 75)(9 150 19 160)(10 67 20 77)(21 153 31 143)(22 80 32 70)(23 155 33 145)(24 62 34 72)(25 157 35 147)(26 64 36 74)(27 159 37 149)(28 66 38 76)(29 141 39 151)(30 68 40 78)(41 83 51 93)(42 105 52 115)(43 85 53 95)(44 107 54 117)(45 87 55 97)(46 109 56 119)(47 89 57 99)(48 111 58 101)(49 91 59 81)(50 113 60 103)(82 139 92 129)(84 121 94 131)(86 123 96 133)(88 125 98 135)(90 127 100 137)(102 128 112 138)(104 130 114 140)(106 132 116 122)(108 134 118 124)(110 136 120 126)
(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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 85)(22 84)(23 83)(24 82)(25 81)(26 100)(27 99)(28 98)(29 97)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(37 89)(38 88)(39 87)(40 86)(41 61)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(121 144)(122 143)(123 142)(124 141)(125 160)(126 159)(127 158)(128 157)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)

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

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

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

56 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 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H 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 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 20 20 20 20 2 2 4 4 4 4 5 5 5 5 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C4○D8 D4×D5 D4×D5 Q8.D10 kernel C2×Q8.D10 D5×C2×C8 C2×D40 Q8.D10 C2×Q8⋊D5 C10×Q16 C2×Q8⋊2D5 C4×D5 C2×Dic5 C22×D5 C2×Q16 C2×C8 Q16 C2×Q8 C10 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 2 2 8 4 8 2 2 8

Matrix representation of C2×Q8.D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 32 0 0 0 0 9
,
 6 35 0 0 6 1 0 0 0 0 15 15 0 0 15 26
,
 0 40 0 0 40 0 0 0 0 0 12 29 0 0 29 29
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[6,6,0,0,35,1,0,0,0,0,15,15,0,0,15,26],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29] >;

C2×Q8.D10 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_{10}
% in TeX

G:=Group("C2xQ8.D10");
// GroupNames label

G:=SmallGroup(320,1437);
// by ID

G=gap.SmallGroup(320,1437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,1123,185,136,438,235,102,12550]);
// Polycyclic

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

׿
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𝔽