C5xC2^3.4Q8 - GroupNames

G = C₅×C₂³.₄Q₈ order 320 = 2⁶·5

Direct product of C₅ and C₂³.₄Q₈

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

Aliases: C₅×C₂³.₄Q₈, (C₂×C₂₀).₃₁₂D₄, C₂³.₄(C₅×Q₈), C₂⁴.₉(C₂×C₁₀), (C₂²×C₁₀).₄Q₈, C₂².₇₅(D₄×C₁₀), C₂².₂₄(Q₈×C₁₀), C₁₀.₄₁(C₄⋊₁D₄), C₁₀.₉₁(C₂²⋊Q₈), (C₂³×C₁₀).₉C₂², C₂.C₄²⋊₁₂C₁₀, C₂³.₈₂(C₂²×C₁₀), (C₂²×C₂₀).₃₆C₂², (C₂²×C₁₀).₄₆₃C₂³, C₁₀.₉₃(C₂².D₄), (C₂×C₄⋊C₄)⋊₈C₁₀, (C₁₀×C₄⋊C₄)⋊₃₅C₂, (C₂×C₄).₁₉(C₅×D₄), C₂.₄(C₅×C₄⋊₁D₄), C₂.₁₀(C₅×C₂²⋊Q₈), (C₂×C₁₀).₆₁₅(C₂×D₄), (C₂×C₁₀).₁₁₂(C₂×Q₈), C₂².₄₂(C₅×C₄○D₄), (C₁₀×C₂²⋊C₄).₃₀C₂, (C₂×C₂²⋊C₄).₁₀C₁₀, (C₂²×C₄).₂₃(C₂×C₁₀), (C₂×C₁₀).₂₂₃(C₄○D₄), C₂.₉(C₅×C₂².D₄), (C₅×C₂.C₄²)⋊₂₈C₂, SmallGroup(320,900)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₅×C₂³.₄Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₁₀ — C₂²×C₂₀ — C₁₀×C₄⋊C₄ — C₅×C₂³.₄Q₈

Lower central

C₁ — C₂³ — C₅×C₂³.₄Q₈

Upper central

C₁ — C₂²×C₁₀ — C₅×C₂³.₄Q₈

Generators and relations for C₅×C₂³.₄Q₈
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e⁴=1, f²=ce², ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=ce^-1 >

Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³.₄Q₈, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂²×C₂₀, C₂³×C₁₀, C₅×C₂.C₄², C₁₀×C₂²⋊C₄, C₁₀×C₄⋊C₄, C₅×C₂³.₄Q₈
Quotients: C₁, C₂, C₂², C₅, D₄, Q₈, C₂³, C₁₀, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₁₀, C₂²⋊Q₈, C₂².D₄, C₄⋊₁D₄, C₅×D₄, C₅×Q₈, C₂²×C₁₀, C₂³.₄Q₈, D₄×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₅×C₂²⋊Q₈, C₅×C₂².D₄, C₅×C₄⋊₁D₄, C₅×C₂³.₄Q₈

Smallest permutation representation of C₅×C₂³.₄Q₈
►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 45)(2 41)(3 42)(4 43)(5 44)(6 31)(7 32)(8 33)(9 34)(10 35)(11 53)(12 54)(13 55)(14 51)(15 52)(16 25)(17 21)(18 22)(19 23)(20 24)(26 69)(27 70)(28 66)(29 67)(30 68)(36 46)(37 47)(38 48)(39 49)(40 50)(56 65)(57 61)(58 62)(59 63)(60 64)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 146)(97 147)(98 148)(99 149)(100 150)(101 128)(102 129)(103 130)(104 126)(105 127)(111 124)(112 125)(113 121)(114 122)(115 123)(116 133)(117 134)(118 135)(119 131)(120 132)(136 157)(137 158)(138 159)(139 160)(140 156)(141 154)(142 155)(143 151)(144 152)(145 153)

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

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

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

(1 96 45 121)(2 97 41 122)(3 98 42 123)(4 99 43 124)(5 100 44 125)(6 85 140 60)(7 81 136 56)(8 82 137 57)(9 83 138 58)(10 84 139 59)(11 133 53 104)(12 134 54 105)(13 135 55 101)(14 131 51 102)(15 132 52 103)(16 95 141 61)(17 91 142 62)(18 92 143 63)(19 93 144 64)(20 94 145 65)(21 108 155 75)(22 109 151 71)(23 110 152 72)(24 106 153 73)(25 107 154 74)(26 149 69 111)(27 150 70 112)(28 146 66 113)(29 147 67 114)(30 148 68 115)(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)

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

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

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

110 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4L	5A	5B	5C	5D	10A	···	10AB	10AC	···	10AJ	20A	···	20AV
order	1	2	···	2	2	2	4	···	4	5	5	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	4	4	4	···	4	1	1	1	1	1	···	1	4	···	4	4	···	4

110 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+					+	-
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	Q₈	C₄○D₄	C₅×D₄	C₅×Q₈	C₅×C₄○D₄
kernel	C₅×C₂³.₄Q₈	C₅×C₂.C₄²	C₁₀×C₂²⋊C₄	C₁₀×C₄⋊C₄	C₂³.₄Q₈	C₂.C₄²	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₂×C₂₀	C₂²×C₁₀	C₂×C₁₀	C₂×C₄	C₂³	C₂²
# reps	1	1	3	3	4	4	12	12	6	2	6	24	8	24

Matrix representation of C₅×C₂³.₄Q₈ ►in GL₆(𝔽₄₁)

37	0	0	0	0	0
0	37	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	37	0
0	0	0	0	0	37

1	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	1

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

0	1	0	0	0	0
40	0	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	9	0	0	0	0
32	0	0	0	0	0
0	0	9	0	0	0
0	0	0	32	0	0
0	0	0	0	0	40
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,40,0] >;

C₅×C₂³.₄Q₈ in GAP, Magma, Sage, TeX

C_5\times C_2^3._4Q_8

% in TeX

G:=Group("C5xC2^3.4Q8");

// GroupNames label

G:=SmallGroup(320,900);

// by ID

G=gap.SmallGroup(320,900);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,848,1766,1731]);

// Polycyclic

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

// generators/relations

G = C5×C23.4Q8 order 320 = 26·5

Direct product of C5 and C23.4Q8

G = C₅×C₂³.₄Q₈ order 320 = 2⁶·5

Direct product of C₅ and C₂³.₄Q₈