Copied to
clipboard

G = C5×D4⋊Q8order 320 = 26·5

Direct product of C5 and D4⋊Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×D4⋊Q8, C20.64D8, C4⋊C84C10, C4⋊Q83C10, (C5×D4)⋊8Q8, D41(C5×Q8), C2.D84C10, C4.13(C5×D8), C2.7(C10×D8), (C4×D4).7C10, C10.79(C2×D8), C4.12(Q8×C10), (D4×C20).22C2, (C2×C20).330D4, C20.118(C2×Q8), D4⋊C4.3C10, C42.20(C2×C10), C22.95(D4×C10), C20.311(C4○D4), (C2×C40).259C22, (C2×C20).930C23, (C4×C20).262C22, C10.93(C22⋊Q8), (D4×C10).298C22, C10.139(C8.C22), (C5×C4⋊C8)⋊23C2, (C5×C4⋊Q8)⋊24C2, (C2×C8).6(C2×C10), (C5×C2.D8)⋊19C2, C4.23(C5×C4○D4), C4⋊C4.11(C2×C10), (C2×C4).131(C5×D4), C2.12(C5×C22⋊Q8), (C2×D4).58(C2×C10), (C2×C10).651(C2×D4), C2.14(C5×C8.C22), (C5×D4⋊C4).12C2, (C5×C4⋊C4).233C22, (C2×C4).105(C22×C10), SmallGroup(320,975)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×D4⋊Q8
C1C2C4C2×C4C2×C20C5×C4⋊C4C5×C4⋊Q8 — C5×D4⋊Q8
C1C2C2×C4 — C5×D4⋊Q8
C1C2×C10C4×C20 — C5×D4⋊Q8

Generators and relations for C5×D4⋊Q8
 G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 202 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C40 [×2], C2×C20 [×3], C2×C20 [×5], C5×D4 [×2], C5×D4, C5×Q8 [×2], C22×C10, D4⋊Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C22×C20, D4×C10, Q8×C10, C5×D4⋊C4 [×2], C5×C4⋊C8, C5×C2.D8 [×2], D4×C20, C5×C4⋊Q8, C5×D4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], C22⋊Q8, C2×D8, C8.C22, C5×D4 [×2], C5×Q8 [×2], C22×C10, D4⋊Q8, C5×D8 [×2], D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C10×D8, C5×C8.C22, C5×D4⋊Q8

Smallest permutation representation of C5×D4⋊Q8
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 37 29 66)(2 38 30 67)(3 39 26 68)(4 40 27 69)(5 36 28 70)(6 20 136 156)(7 16 137 157)(8 17 138 158)(9 18 139 159)(10 19 140 160)(11 46 53 41)(12 47 54 42)(13 48 55 43)(14 49 51 44)(15 50 52 45)(21 35 142 151)(22 31 143 152)(23 32 144 153)(24 33 145 154)(25 34 141 155)(56 110 85 76)(57 106 81 77)(58 107 82 78)(59 108 83 79)(60 109 84 80)(61 89 91 75)(62 90 92 71)(63 86 93 72)(64 87 94 73)(65 88 95 74)(96 116 125 150)(97 117 121 146)(98 118 122 147)(99 119 123 148)(100 120 124 149)(101 115 131 129)(102 111 132 130)(103 112 133 126)(104 113 134 127)(105 114 135 128)
(1 42)(2 43)(3 44)(4 45)(5 41)(6 22)(7 23)(8 24)(9 25)(10 21)(11 70)(12 66)(13 67)(14 68)(15 69)(16 153)(17 154)(18 155)(19 151)(20 152)(26 49)(27 50)(28 46)(29 47)(30 48)(31 156)(32 157)(33 158)(34 159)(35 160)(36 53)(37 54)(38 55)(39 51)(40 52)(56 86)(57 87)(58 88)(59 89)(60 90)(61 108)(62 109)(63 110)(64 106)(65 107)(71 84)(72 85)(73 81)(74 82)(75 83)(76 93)(77 94)(78 95)(79 91)(80 92)(96 103)(97 104)(98 105)(99 101)(100 102)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)(121 134)(122 135)(123 131)(124 132)(125 133)(136 143)(137 144)(138 145)(139 141)(140 142)
(1 64 12 81)(2 65 13 82)(3 61 14 83)(4 62 15 84)(5 63 11 85)(6 103 143 125)(7 104 144 121)(8 105 145 122)(9 101 141 123)(10 102 142 124)(16 113 153 146)(17 114 154 147)(18 115 155 148)(19 111 151 149)(20 112 152 150)(21 100 140 132)(22 96 136 133)(23 97 137 134)(24 98 138 135)(25 99 139 131)(26 91 51 59)(27 92 52 60)(28 93 53 56)(29 94 54 57)(30 95 55 58)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 110 70 72)(42 106 66 73)(43 107 67 74)(44 108 68 75)(45 109 69 71)
(1 104 12 121)(2 105 13 122)(3 101 14 123)(4 102 15 124)(5 103 11 125)(6 85 143 63)(7 81 144 64)(8 82 145 65)(9 83 141 61)(10 84 142 62)(16 106 153 73)(17 107 154 74)(18 108 155 75)(19 109 151 71)(20 110 152 72)(21 92 140 60)(22 93 136 56)(23 94 137 57)(24 95 138 58)(25 91 139 59)(26 131 51 99)(27 132 52 100)(28 133 53 96)(29 134 54 97)(30 135 55 98)(31 86 156 76)(32 87 157 77)(33 88 158 78)(34 89 159 79)(35 90 160 80)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(41 150 70 112)(42 146 66 113)(43 147 67 114)(44 148 68 115)(45 149 69 111)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20P20Q···20AB20AC···20AJ40A···40P
order1222224444444445555888810···1010···1020···2020···2020···2040···40
size111144222244488111144441···14···42···24···48···84···4

95 irreducible representations

dim1111111111112222222244
type+++++++-+-
imageC1C2C2C2C2C2C5C10C10C10C10C10D4Q8D8C4○D4C5×D4C5×Q8C5×D8C5×C4○D4C8.C22C5×C8.C22
kernelC5×D4⋊Q8C5×D4⋊C4C5×C4⋊C8C5×C2.D8D4×C20C5×C4⋊Q8D4⋊Q8D4⋊C4C4⋊C8C2.D8C4×D4C4⋊Q8C2×C20C5×D4C20C20C2×C4D4C4C4C10C2
# reps12121148484422428816814

Matrix representation of C5×D4⋊Q8 in GL4(𝔽41) generated by

37000
03700
00100
00010
,
40000
04000
0001
00400
,
1000
04000
0001
0010
,
9000
03200
0010
0001
,
0100
40000
001229
002929
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[9,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,12,29,0,0,29,29] >;

C5×D4⋊Q8 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes Q_8
% in TeX

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

G:=SmallGroup(320,975);
// by ID

G=gap.SmallGroup(320,975);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,10085,2539,124]);
// 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=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽