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G = Q82D5⋊C4order 320 = 26·5

2nd semidirect product of Q82D5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82D52C4, (C4×D5).96D4, C4.165(D4×D5), Q8.10(C4×D5), C4⋊C4.150D10, Q8⋊C422D5, D20.17(C2×C4), C20.119(C2×D4), Q8⋊Dic56C2, (C2×C8).209D10, D205C423C2, D206C410C2, C22.79(D4×D5), C10.47(C4○D8), C20.50(C22×C4), (C2×Q8).106D10, (C22×D5).80D4, C2.2(Q8.D10), (C2×C40).199C22, (C2×C20).244C23, (C2×Dic5).272D4, (C2×D20).65C22, C53(C23.24D4), C4⋊Dic5.92C22, (Q8×C10).27C22, D10.26(C22⋊C4), C2.4(SD163D5), Dic5.57(C22⋊C4), (D5×C2×C8)⋊20C2, C4.15(C2×C4×D5), C4⋊C47D55C2, (C4×D5).51(C2×C4), (C5×Q8).19(C2×C4), C2.24(D5×C22⋊C4), (C5×Q8⋊C4)⋊21C2, (C2×C10).257(C2×D4), (C5×C4⋊C4).45C22, C10.64(C2×C22⋊C4), (C2×Q82D5).3C2, (C2×C4×D5).297C22, (C2×C4).351(C22×D5), (C2×C52C8).227C22, SmallGroup(320,431)

Series: Derived Chief Lower central Upper central

C1C20 — Q82D5⋊C4
C1C5C10C20C2×C20C2×C4×D5C2×Q82D5 — Q82D5⋊C4
C5C10C20 — Q82D5⋊C4
C1C22C2×C4Q8⋊C4

Generators and relations for Q82D5⋊C4
 G = < a,b,c,d,e | a4=c5=d2=e4=1, b2=a2, bab-1=dad=eae-1=a-1, ac=ca, bc=cb, bd=db, ebe-1=a-1b, dcd=c-1, ce=ec, ede-1=a-1d >

Subgroups: 654 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×12], D4 [×7], Q8 [×2], Q8, C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8 [×3], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×6], C2×C10, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C52C8, C40, C4×D5 [×4], C4×D5 [×4], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C23.24D4, C8×D5 [×2], C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5 [×4], Q82D5 [×2], Q8×C10, D206C4, D205C4, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C47D5, D5×C2×C8, C2×Q82D5, Q82D5⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4○D8 [×2], C4×D5 [×2], C22×D5, C23.24D4, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, SD163D5, Q8.D10, Q82D5⋊C4

Smallest permutation representation of Q82D5⋊C4
On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 91 6 96)(2 92 7 97)(3 93 8 98)(4 94 9 99)(5 95 10 100)(11 86 16 81)(12 87 17 82)(13 88 18 83)(14 89 19 84)(15 90 20 85)(21 111 26 116)(22 112 27 117)(23 113 28 118)(24 114 29 119)(25 115 30 120)(31 106 36 101)(32 107 37 102)(33 108 38 103)(34 109 39 104)(35 110 40 105)(41 131 46 136)(42 132 47 137)(43 133 48 138)(44 134 49 139)(45 135 50 140)(51 126 56 121)(52 127 57 122)(53 128 58 123)(54 129 59 124)(55 130 60 125)(61 151 66 156)(62 152 67 157)(63 153 68 158)(64 154 69 159)(65 155 70 160)(71 146 76 141)(72 147 77 142)(73 148 78 143)(74 149 79 144)(75 150 80 145)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 85)(82 84)(86 90)(87 89)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 115)(112 114)(116 120)(117 119)(121 135)(122 134)(123 133)(124 132)(125 131)(126 140)(127 139)(128 138)(129 137)(130 136)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
(1 156 36 121)(2 157 37 122)(3 158 38 123)(4 159 39 124)(5 160 40 125)(6 151 31 126)(7 152 32 127)(8 153 33 128)(9 154 34 129)(10 155 35 130)(11 141 21 131)(12 142 22 132)(13 143 23 133)(14 144 24 134)(15 145 25 135)(16 146 26 136)(17 147 27 137)(18 148 28 138)(19 149 29 139)(20 150 30 140)(41 91 76 101)(42 92 77 102)(43 93 78 103)(44 94 79 104)(45 95 80 105)(46 96 71 106)(47 97 72 107)(48 98 73 108)(49 99 74 109)(50 100 75 110)(51 86 61 111)(52 87 62 112)(53 88 63 113)(54 89 64 114)(55 90 65 115)(56 81 66 116)(57 82 67 117)(58 83 68 118)(59 84 69 119)(60 85 70 120)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444444444558888888810···102020202020···2040···40
size11111010202022444455552020222222101010102···244448···84···4

56 irreducible representations

dim1111111112222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4D4D5D10D10D10C4○D8C4×D5D4×D5D4×D5SD163D5Q8.D10
kernelQ82D5⋊C4D206C4D205C4Q8⋊Dic5C5×Q8⋊C4C4⋊C47D5D5×C2×C8C2×Q82D5Q82D5C4×D5C2×Dic5C22×D5Q8⋊C4C4⋊C4C2×C8C2×Q8C10Q8C4C22C2C2
# reps1111111182112222882244

Matrix representation of Q82D5⋊C4 in GL4(𝔽41) generated by

1000
0100
0001
00400
,
1000
0100
00032
00320
,
64000
1000
0010
0001
,
64000
353500
0001
0010
,
9000
0900
002626
002615
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,32,0],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,0,1,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,26,26,0,0,26,15] >;

Q82D5⋊C4 in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_5\rtimes C_4
% in TeX

G:=Group("Q8:2D5:C4");
// GroupNames label

G:=SmallGroup(320,431);
// by ID

G=gap.SmallGroup(320,431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,219,58,570,136,851,102,12550]);
// Polycyclic

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

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