D20.35C2^3 - GroupNames

G = D₂₀.₃₅C₂³ order 320 = 2⁶·5

16^th non-split extension by D₂₀ of C₂³ acting via C₂³/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₀.₄₀C₂⁴, D₂₀.₃₅C₂³, 2_-¹⁺⁴⋊₄D₅, Dic₁₀.₃₅C₂³, C₅⋊₅(Q₈○D₈), (C₅×D₄).₃₉D₄, (C₅×Q₈).₃₉D₄, C₄○D₄.₁₇D₁₀, C₂₀.₂₇₂(C₂×D₄), D₄⋊D₅.₂C₂², (C₂×Q₈).₉₃D₁₀, C₄.₄₀(C₂³×D₅), Q₈⋊D₅.₃C₂², D₄.₂₁(C₅⋊D₄), D₄.Dic₅⋊₁₃C₂, C₅⋊₂C₈.₁₉C₂³, Q₈.₂₁(C₅⋊D₄), (C₅×D₄).₂₈C₂³, D₄.₂₈(C₂²×D₅), D₄.D₅.₃C₂², D₄.₉D₁₀⋊₁₂C₂, D₄.₈D₁₀⋊₁₁C₂, (C₅×Q₈).₂₈C₂³, Q₈.₂₈(C₂²×D₅), C₂₀.C₂³⋊₁₂C₂, C₅⋊Q₁₆.₄C₂², (C₂×C₂₀).₁₂₁C₂³, C₄○D₂₀.₃₄C₂², C₁₀.₁₇₄(C₂²×D₄), (C₅×2_-¹⁺⁴)⋊₃C₂, D₄.₁₀D₁₀⋊₁₀C₂, (Q₈×C₁₀).₁₅₄C₂², C₄.Dic₅.₃₂C₂², (C₂×Dic₁₀).₂₁₁C₂², C₄.₇₈(C₂×C₅⋊D₄), (C₂×C₅⋊Q₁₆)⋊₃₂C₂, (C₂×C₁₀).₈₈(C₂×D₄), C₂².₉(C₂×C₅⋊D₄), C₂.₄₇(C₂²×C₅⋊D₄), (C₅×C₄○D₄).₃₀C₂², (C₂×C₄).₁₀₅(C₂²×D₅), (C₂×C₅⋊₂C₈).₁₈₅C₂², SmallGroup(320,1510)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₂₀.₃₅C₂³

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — D₂₀ — C₄○D₂₀ — D₄.₁₀D₁₀ — D₂₀.₃₅C₂³

Lower central

C₅ — C₁₀ — C₂₀ — D₂₀.₃₅C₂³

Upper central

C₁ — C₂ — C₄○D₄ — 2_-¹⁺⁴

Generators and relations for D₂₀.₃₅C₂³
G = < a,b,c,d,e | a²⁰=b²=e²=1, c²=d²=a¹⁰, bab=a^-1, ac=ca, ad=da, eae=a¹¹, bc=cb, bd=db, ebe=a¹⁵b, dcd^-1=a¹⁰c, ce=ec, de=ed >

Subgroups: 726 in 248 conjugacy classes, 107 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, D₄, Q₈, Q₈, Q₈, D₅, C₁₀, C₁₀, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₄○D₄, Dic₅, C₂₀, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₈○D₄, C₂×Q₁₆, C₄○D₈, C₈.C₂², 2_-¹⁺⁴, 2_-¹⁺⁴, C₅⋊₂C₈, C₅⋊₂C₈, Dic₁₀, Dic₁₀, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×Q₈, C₅×Q₈, Q₈○D₈, C₂×C₅⋊₂C₈, C₄.Dic₅, D₄⋊D₅, D₄.D₅, Q₈⋊D₅, C₅⋊Q₁₆, C₂×Dic₁₀, C₄○D₂₀, D₄⋊₂D₅, Q₈×D₅, Q₈×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₅×C₄○D₄, C₂₀.C₂³, C₂×C₅⋊Q₁₆, D₄.Dic₅, D₄.₈D₁₀, D₄.₉D₁₀, D₄.₁₀D₁₀, C₅×2_-¹⁺⁴, D₂₀.₃₅C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, C₅⋊D₄, C₂²×D₅, Q₈○D₈, C₂×C₅⋊D₄, C₂³×D₅, C₂²×C₅⋊D₄, D₂₀.₃₅C₂³

Smallest permutation representation of D₂₀.₃₅C₂³
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 140)(138 139)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	5A	5B	8A	8B	8C	8D	8E	10A	10B	10C	···	10L	20A	···	20T
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	5	5	8	8	8	8	8	10	10	10	···	10	20	···	20
size	1	1	2	2	2	4	20	2	2	2	2	4	4	4	20	20	20	2	2	10	10	20	20	20	2	2	4	···	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+			-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₅	D₁₀	D₁₀	C₅⋊D₄	C₅⋊D₄	Q₈○D₈	D₂₀.₃₅C₂³
kernel	D₂₀.₃₅C₂³	C₂₀.C₂³	C₂×C₅⋊Q₁₆	D₄.Dic₅	D₄.₈D₁₀	D₄.₉D₁₀	D₄.₁₀D₁₀	C₅×2_-¹⁺⁴	C₅×D₄	C₅×Q₈	2_-¹⁺⁴	C₂×Q₈	C₄○D₄	D₄	Q₈	C₅	C₁
# reps	1	3	3	1	3	3	1	1	3	1	2	6	8	12	4	2	2

Matrix representation of D₂₀.₃₅C₂³ ►in GL₆(𝔽₄₁)

34	1	0	0	0	0
40	0	0	0	0	0
0	0	1	21	0	0
0	0	37	40	0	0
0	0	10	33	1	18
0	0	29	2	9	40

1	34	0	0	0	0
0	40	0	0	0	0
0	0	1	21	0	0
0	0	0	40	0	0
0	0	0	33	1	18
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	14	12	13	0
0	0	0	4	0	17
0	0	29	37	27	19
0	0	0	40	0	37

40	0	0	0	0	0
0	40	0	0	0	0
0	0	30	2	9	0
0	0	0	1	0	37
0	0	32	25	11	10
0	0	0	21	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	24	6	0	0
0	0	34	17	0	0
0	0	38	5	24	11
0	0	20	24	26	17

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,37,10,29,0,0,21,40,33,2,0,0,0,0,1,9,0,0,0,0,18,40],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,33,0,0,0,0,0,1,0,0,0,0,0,18,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,29,0,0,0,12,4,37,40,0,0,13,0,27,0,0,0,0,17,19,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,0,32,0,0,0,2,1,25,21,0,0,9,0,11,0,0,0,0,37,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,34,38,20,0,0,6,17,5,24,0,0,0,0,24,26,0,0,0,0,11,17] >;

D₂₀.₃₅C₂³ in GAP, Magma, Sage, TeX

D_{20}._{35}C_2^3

% in TeX

G:=Group("D20.35C2^3");

// GroupNames label

G:=SmallGroup(320,1510);

// by ID

G=gap.SmallGroup(320,1510);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,1684,235,102,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^20=b^2=e^2=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^15*b,d*c*d^-1=a^10*c,c*e=e*c,d*e=e*d>;

// generators/relations

G = D20.35C23 order 320 = 26·5

16th non-split extension by D20 of C23 acting via C23/C22=C2

G = D₂₀.₃₅C₂³ order 320 = 2⁶·5

16^th non-split extension by D₂₀ of C₂³ acting via C₂³/C₂²=C₂