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## G = C2×Q8.10D10order 320 = 26·5

### Direct product of C2 and Q8.10D10

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×Q8.10D10
G = < a,b,c,d,e | a2=b4=1, c2=d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d9 >

Subgroups: 2190 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×Q8, C22×Q8, C2×C4○D4, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×Q8, C22×D5, C22×C10, C2×2- 1+4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, Q8×D5, Q82D5, C2×C5⋊D4, C22×C20, Q8×C10, C2×C4○D20, C2×Q8×D5, C2×Q82D5, Q8.10D10, Q8×C2×C10, C2×Q8.10D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C25, C22×D5, C2×2- 1+4, C23×D5, Q8.10D10, D5×C24, C2×Q8.10D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 4A ··· 4L 4M ··· 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D5 D10 D10 2- 1+4 Q8.10D10 kernel C2×Q8.10D10 C2×C4○D20 C2×Q8×D5 C2×Q8⋊2D5 Q8.10D10 Q8×C2×C10 C22×Q8 C22×C4 C2×Q8 C10 C2 # reps 1 6 4 4 16 1 2 6 24 2 8

Matrix representation of C2×Q8.10D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 40 27 0 0 0 8 37 0 27 0 0 27 0 37 1 0 0 0 27 33 4
,
 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
,
 1 35 0 0 0 0 6 6 0 0 0 0 0 0 33 1 1 34 0 0 33 0 15 27 0 0 1 34 8 40 0 0 15 27 8 0
,
 1 35 0 0 0 0 0 40 0 0 0 0 0 0 24 23 0 0 0 0 7 17 0 0 0 0 0 0 17 18 0 0 0 0 34 24

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,8,27,0,0,0,40,37,0,27,0,0,27,0,37,33,0,0,0,27,1,4],[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],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,33,33,1,15,0,0,1,0,34,27,0,0,1,15,8,8,0,0,34,27,40,0],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,24,7,0,0,0,0,23,17,0,0,0,0,0,0,17,34,0,0,0,0,18,24] >;

C2×Q8.10D10 in GAP, Magma, Sage, TeX

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

G:=Group("C2xQ8.10D10");
// GroupNames label

G:=SmallGroup(320,1617);
// by ID

G=gap.SmallGroup(320,1617);
# by ID

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

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

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