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## G = C2×D10⋊2Q8order 320 = 26·5

### Direct product of C2 and D10⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D10⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10⋊2Q8
 Lower central C5 — C2×C10 — C2×D10⋊2Q8
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×D102Q8
G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 1134 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×8], C24, Dic5 [×6], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C4⋊Dic5 [×8], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, D102Q8 [×8], C2×C4⋊Dic5 [×2], C2×D10⋊C4 [×2], C10×C4⋊C4, C22×Dic10, D5×C22×C4, C2×D102Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, D20 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×D20 [×6], D42D5 [×2], Q8×D5 [×2], C23×D5, D102Q8 [×4], C22×D20, C2×D42D5, C2×Q8×D5, C2×D102Q8

Smallest permutation representation of C2×D102Q8
On 160 points
Generators in S160
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 146)(52 147)(53 148)(54 149)(55 150)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(81 156)(82 157)(83 158)(84 159)(85 160)(86 151)(87 152)(88 153)(89 154)(90 155)
(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)
(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 74)(12 73)(13 72)(14 71)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 93)(22 92)(23 91)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 130)(40 129)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 120)(50 119)(51 143)(52 142)(53 141)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 133)(62 132)(63 131)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 160)(90 159)
(1 115 25 130)(2 114 26 129)(3 113 27 128)(4 112 28 127)(5 111 29 126)(6 120 30 125)(7 119 21 124)(8 118 22 123)(9 117 23 122)(10 116 24 121)(11 70 160 55)(12 69 151 54)(13 68 152 53)(14 67 153 52)(15 66 154 51)(16 65 155 60)(17 64 156 59)(18 63 157 58)(19 62 158 57)(20 61 159 56)(31 109 46 94)(32 108 47 93)(33 107 48 92)(34 106 49 91)(35 105 50 100)(36 104 41 99)(37 103 42 98)(38 102 43 97)(39 101 44 96)(40 110 45 95)(71 134 86 149)(72 133 87 148)(73 132 88 147)(74 131 89 146)(75 140 90 145)(76 139 81 144)(77 138 82 143)(78 137 83 142)(79 136 84 141)(80 135 85 150)
(1 65 25 60)(2 66 26 51)(3 67 27 52)(4 68 28 53)(5 69 29 54)(6 70 30 55)(7 61 21 56)(8 62 22 57)(9 63 23 58)(10 64 24 59)(11 125 160 120)(12 126 151 111)(13 127 152 112)(14 128 153 113)(15 129 154 114)(16 130 155 115)(17 121 156 116)(18 122 157 117)(19 123 158 118)(20 124 159 119)(31 86 46 71)(32 87 47 72)(33 88 48 73)(34 89 49 74)(35 90 50 75)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(91 146 106 131)(92 147 107 132)(93 148 108 133)(94 149 109 134)(95 150 110 135)(96 141 101 136)(97 142 102 137)(98 143 103 138)(99 144 104 139)(100 145 105 140)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 D20 D4⋊2D5 Q8×D5 kernel C2×D10⋊2Q8 D10⋊2Q8 C2×C4⋊Dic5 C2×D10⋊C4 C10×C4⋊C4 C22×Dic10 D5×C22×C4 C2×C20 C22×D5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 2 2 1 1 1 4 4 2 4 8 6 16 4 4

Matrix representation of C2×D102Q8 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 7 0 0 0 35 35
,
 1 0 0 0 0 0 40 0 0 0 0 15 1 0 0 0 0 0 40 0 0 0 0 36 1
,
 40 0 0 0 0 0 17 5 0 0 0 24 24 0 0 0 0 0 25 14 0 0 0 14 16
,
 40 0 0 0 0 0 32 0 0 0 0 12 9 0 0 0 0 0 40 0 0 0 0 0 40

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,35,0,0,0,7,35],[1,0,0,0,0,0,40,15,0,0,0,0,1,0,0,0,0,0,40,36,0,0,0,0,1],[40,0,0,0,0,0,17,24,0,0,0,5,24,0,0,0,0,0,25,14,0,0,0,14,16],[40,0,0,0,0,0,32,12,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40] >;

C2×D102Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes_2Q_8
% in TeX

G:=Group("C2xD10:2Q8");
// GroupNames label

G:=SmallGroup(320,1181);
// by ID

G=gap.SmallGroup(320,1181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,192,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
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