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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 [×2], C2 [×10], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×18], C5, C2×C4 [×18], C2×C4 [×52], D4 [×40], Q8 [×16], Q8 [×24], C23, C23 [×4], D5 [×8], C10, C10 [×2], C10 [×2], C22×C4 [×3], C22×C4 [×12], C2×D4 [×10], C2×Q8 [×12], C2×Q8 [×38], C4○D4 [×80], Dic5 [×8], C20 [×12], D10 [×8], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×Q8, C22×Q8 [×4], C2×C4○D4 [×10], 2- 1+4 [×16], Dic10 [×24], C4×D5 [×48], D20 [×24], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×18], C5×Q8 [×16], C22×D5 [×4], C22×C10, C2×2- 1+4, C2×Dic10 [×6], C2×C4×D5 [×12], C2×D20 [×6], C4○D20 [×48], Q8×D5 [×32], Q82D5 [×32], C2×C5⋊D4 [×4], C22×C20 [×3], Q8×C10 [×12], C2×C4○D20 [×6], C2×Q8×D5 [×4], C2×Q82D5 [×4], Q8.10D10 [×16], Q8×C2×C10, C2×Q8.10D10
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2- 1+4 [×2], C25, C22×D5 [×35], C2×2- 1+4, C23×D5 [×15], Q8.10D10 [×2], D5×C24, C2×Q8.10D10

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

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

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

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

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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