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G = C5×D43Q8order 320 = 26·5

Direct product of C5 and D43Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×D43Q8, C10.1652+ 1+4, D43(C5×Q8), C4⋊Q815C10, (C5×D4)⋊10Q8, (C4×Q8)⋊15C10, (Q8×C20)⋊35C2, C4.18(Q8×C10), (C4×D4).12C10, (D4×C20).27C2, C22⋊Q817C10, C20.124(C2×Q8), C22.6(Q8×C10), C42.48(C2×C10), C42.C210C10, C20.325(C4○D4), C10.64(C22×Q8), (C2×C20).680C23, (C2×C10).374C24, (C4×C20).289C22, (D4×C10).335C22, C22.48(C23×C10), C23.45(C22×C10), (Q8×C10).184C22, C2.17(C5×2+ 1+4), (C22×C20).459C22, (C22×C10).268C23, (C5×C4⋊Q8)⋊36C2, (C2×C4⋊C4)⋊23C10, (C10×C4⋊C4)⋊50C2, C2.10(Q8×C2×C10), C4.37(C5×C4○D4), C4⋊C4.74(C2×C10), C2.27(C10×C4○D4), (C5×C22⋊Q8)⋊44C2, (C2×C10).55(C2×Q8), (C2×D4).81(C2×C10), C10.246(C2×C4○D4), (C2×Q8).64(C2×C10), (C5×C42.C2)⋊27C2, (C5×C4⋊C4).400C22, C22⋊C4.24(C2×C10), (C2×C4).36(C22×C10), (C22×C4).70(C2×C10), (C5×C22⋊C4).156C22, SmallGroup(320,1556)

Series: Derived Chief Lower central Upper central

C1C22 — C5×D43Q8
C1C2C22C2×C10C2×C20C5×C4⋊C4C5×C22⋊Q8 — C5×D43Q8
C1C22 — C5×D43Q8
C1C2×C10 — C5×D43Q8

Generators and relations for C5×D43Q8
 G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >

Subgroups: 314 in 228 conjugacy classes, 166 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×10], C2×C4 [×8], D4 [×4], Q8 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], C20 [×4], C20 [×11], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, C2×C20 [×3], C2×C20 [×10], C2×C20 [×8], C5×D4 [×4], C5×Q8 [×4], C22×C10 [×2], D43Q8, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×6], C5×C4⋊C4 [×2], C5×C4⋊C4 [×14], C22×C20 [×6], D4×C10, Q8×C10, Q8×C10 [×2], C10×C4⋊C4 [×2], D4×C20, D4×C20 [×2], Q8×C20, C5×C22⋊Q8 [×6], C5×C42.C2 [×2], C5×C4⋊Q8, C5×D43Q8
Quotients: C1, C2 [×15], C22 [×35], C5, Q8 [×4], C23 [×15], C10 [×15], C2×Q8 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×Q8, C2×C4○D4, 2+ 1+4, C5×Q8 [×4], C22×C10 [×15], D43Q8, Q8×C10 [×6], C5×C4○D4 [×2], C23×C10, Q8×C2×C10, C10×C4○D4, C5×2+ 1+4, C5×D43Q8

Smallest permutation representation of C5×D43Q8
On 160 points
Generators in S160
(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)
(1 127 21 135)(2 128 22 131)(3 129 23 132)(4 130 24 133)(5 126 25 134)(6 74 20 66)(7 75 16 67)(8 71 17 68)(9 72 18 69)(10 73 19 70)(11 62 157 57)(12 63 158 58)(13 64 159 59)(14 65 160 60)(15 61 156 56)(26 120 34 125)(27 116 35 121)(28 117 31 122)(29 118 32 123)(30 119 33 124)(36 101 41 96)(37 102 42 97)(38 103 43 98)(39 104 44 99)(40 105 45 100)(46 114 54 106)(47 115 55 107)(48 111 51 108)(49 112 52 109)(50 113 53 110)(76 147 81 155)(77 148 82 151)(78 149 83 152)(79 150 84 153)(80 146 85 154)(86 140 94 145)(87 136 95 141)(88 137 91 142)(89 138 92 143)(90 139 93 144)
(1 135)(2 131)(3 132)(4 133)(5 134)(6 74)(7 75)(8 71)(9 72)(10 73)(11 62)(12 63)(13 64)(14 65)(15 61)(16 67)(17 68)(18 69)(19 70)(20 66)(21 127)(22 128)(23 129)(24 130)(25 126)(26 125)(27 121)(28 122)(29 123)(30 124)(31 117)(32 118)(33 119)(34 120)(35 116)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 156)(57 157)(58 158)(59 159)(60 160)(76 155)(77 151)(78 152)(79 153)(80 154)(81 147)(82 148)(83 149)(84 150)(85 146)(86 145)(87 141)(88 142)(89 143)(90 144)(91 137)(92 138)(93 139)(94 140)(95 136)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 76 75 95)(57 77 71 91)(58 78 72 92)(59 79 73 93)(60 80 74 94)(61 81 67 87)(62 82 68 88)(63 83 69 89)(64 84 70 90)(65 85 66 86)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 127 107 121)(102 128 108 122)(103 129 109 123)(104 130 110 124)(105 126 106 125)
(1 7 35 156)(2 8 31 157)(3 9 32 158)(4 10 33 159)(5 6 34 160)(11 22 17 28)(12 23 18 29)(13 24 19 30)(14 25 20 26)(15 21 16 27)(36 136 55 155)(37 137 51 151)(38 138 52 152)(39 139 53 153)(40 140 54 154)(41 141 47 147)(42 142 48 148)(43 143 49 149)(44 144 50 150)(45 145 46 146)(56 127 75 121)(57 128 71 122)(58 129 72 123)(59 130 73 124)(60 126 74 125)(61 135 67 116)(62 131 68 117)(63 132 69 118)(64 133 70 119)(65 134 66 120)(76 101 95 107)(77 102 91 108)(78 103 92 109)(79 104 93 110)(80 105 94 106)(81 96 87 115)(82 97 88 111)(83 98 89 112)(84 99 90 113)(85 100 86 114)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4Q5A5B5C5D10A···10L10M···10AB20A···20AF20AG···20BP
order122222224···44···4555510···1010···1020···2020···20
size111122222···24···411111···12···22···24···4

125 irreducible representations

dim11111111111111222244
type+++++++-+
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10Q8C4○D4C5×Q8C5×C4○D42+ 1+4C5×2+ 1+4
kernelC5×D43Q8C10×C4⋊C4D4×C20Q8×C20C5×C22⋊Q8C5×C42.C2C5×C4⋊Q8D43Q8C2×C4⋊C4C4×D4C4×Q8C22⋊Q8C42.C2C4⋊Q8C5×D4C20D4C4C10C2
# reps123162148124248444161614

Matrix representation of C5×D43Q8 in GL4(𝔽41) generated by

18000
01800
00160
00016
,
1000
0100
001339
00328
,
1000
0100
001339
00228
,
0100
40000
0010
0001
,
203800
382100
003523
00276
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,13,3,0,0,39,28],[1,0,0,0,0,1,0,0,0,0,13,2,0,0,39,28],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[20,38,0,0,38,21,0,0,0,0,35,27,0,0,23,6] >;

C5×D43Q8 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_3Q_8
% in TeX

G:=Group("C5xD4:3Q8");
// GroupNames label

G:=SmallGroup(320,1556);
// by ID

G=gap.SmallGroup(320,1556);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1688,3446,1242,304]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations

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