C2^4.9D10 - GroupNames

G = C₂⁴.₉D₁₀ order 320 = 2⁶·5

9^th non-split extension by C₂⁴ of D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₉D₁₀, (C₂×Dic₅).₆₄D₄, (C₂²×C₁₀).₆₄D₄, (C₂²×C₄).₃₀D₁₀, C₂².₂₄₀(D₄×D₅), C₁₀.₈₄(C₄⋊D₄), C₂³.₁₇(C₅⋊D₄), C₅⋊₃(C₂³.₁₁D₄), C₂.₉(Dic₅⋊D₄), C₁₀.₃₃(C₄.₄D₄), C₂².₉₇(C₄○D₂₀), (C₂²×C₂₀).₂₄C₂², (C₂³×C₁₀).₃₅C₂², C₂³.₃₆₉(C₂²×D₅), C₁₀.₁₀C₄²⋊₁₄C₂, C₁₀.₁₅(C₄²⋊₂C₂), C₂².₉₅(D₄⋊₂D₅), (C₂²×C₁₀).₃₂₇C₂³, C₂.₂₁(D₁₀.₁₂D₄), C₂.₂₁(Dic₅.₅D₄), C₁₀.₅₇(C₂².D₄), C₂.₁₃(C₂³.D₁₀), C₂.₇(C₂³.₁₈D₁₀), C₂.₆(C₂³.₂₃D₁₀), (C₂²×Dic₅).₄₁C₂², (C₂×C₂²⋊C₄).₉D₅, (C₂×C₁₀).₄₃₁(C₂×D₄), (C₁₀×C₂²⋊C₄).₈C₂, (C₂×C₁₀.D₄)⋊₁₀C₂, C₂².₁₂₅(C₂×C₅⋊D₄), (C₂×C₂³.D₅).₁₄C₂, (C₂×C₁₀).₁₄₃(C₄○D₄), SmallGroup(320,579)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — C₂⁴.₉D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×Dic₅ — C₂×C₂³.D₅ — C₂⁴.₉D₁₀

Lower central

C₅ — C₂²×C₁₀ — C₂⁴.₉D₁₀

Upper central

C₁ — C₂³ — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.₉D₁₀
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e¹⁰=c, f²=bcd, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=be⁹ >

Subgroups: 566 in 170 conjugacy classes, 57 normal (51 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³.₁₁D₄, C₁₀.D₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂²×Dic₅, C₂²×C₂₀, C₂³×C₁₀, C₁₀.₁₀C₄², C₂×C₁₀.D₄, C₂×C₂³.D₅, C₁₀×C₂²⋊C₄, C₂⁴.₉D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₄○D₄, D₁₀, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₅⋊D₄, C₂²×D₅, C₂³.₁₁D₄, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, C₂×C₅⋊D₄, C₂³.D₁₀, D₁₀.₁₂D₄, Dic₅.₅D₄, C₂³.₂₃D₁₀, C₂³.₁₈D₁₀, Dic₅⋊D₄, C₂⁴.₉D₁₀

Smallest permutation representation of C₂⁴.₉D₁₀
►On 160 points

Generators in S₁₆₀

(2 138)(4 140)(6 122)(8 124)(10 126)(12 128)(14 130)(16 132)(18 134)(20 136)(21 48)(23 50)(25 52)(27 54)(29 56)(31 58)(33 60)(35 42)(37 44)(39 46)(61 105)(62 72)(63 107)(64 74)(65 109)(66 76)(67 111)(68 78)(69 113)(70 80)(71 115)(73 117)(75 119)(77 101)(79 103)(81 91)(82 142)(83 93)(84 144)(85 95)(86 146)(87 97)(88 148)(89 99)(90 150)(92 152)(94 154)(96 156)(98 158)(100 160)(102 112)(104 114)(106 116)(108 118)(110 120)(141 151)(143 153)(145 155)(147 157)(149 159)

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

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

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

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

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

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

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

62 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	4B	4C	4D	4E	···	4L	5A	5B	10A	···	10N	10O	···	10V	20A	···	20P
order	1	2	···	2	2	2	4	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	4	4	4	4	4	4	20	···	20	2	2	2	···	2	4	···	4	4	···	4

62 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+		+	+			+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	D₅	C₄○D₄	D₁₀	D₁₀	C₅⋊D₄	C₄○D₂₀	D₄×D₅	D₄⋊₂D₅
kernel	C₂⁴.₉D₁₀	C₁₀.₁₀C₄²	C₂×C₁₀.D₄	C₂×C₂³.D₅	C₁₀×C₂²⋊C₄	C₂×Dic₅	C₂²×C₁₀	C₂×C₂²⋊C₄	C₂×C₁₀	C₂²×C₄	C₂⁴	C₂³	C₂²	C₂²	C₂²
# reps	1	3	1	2	1	2	2	2	10	4	2	8	16	2	6

Matrix representation of C₂⁴.₉D₁₀ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	30	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	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

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	1	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	1	0
0	0	0	0	0	1

0	1	0	0	0	0
40	0	0	0	0	0
0	0	8	0	0	0
0	0	31	36	0	0
0	0	0	0	31	0
0	0	0	0	0	37

1	0	0	0	0	0
0	1	0	0	0	0
0	0	37	3	0	0
0	0	36	4	0	0
0	0	0	0	0	4
0	0	0	0	10	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,30,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,8,31,0,0,0,0,0,36,0,0,0,0,0,0,31,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,36,0,0,0,0,3,4,0,0,0,0,0,0,0,10,0,0,0,0,4,0] >;

C₂⁴.₉D₁₀ in GAP, Magma, Sage, TeX

C_2^4._9D_{10}

% in TeX

G:=Group("C2^4.9D10");

// GroupNames label

G:=SmallGroup(320,579);

// by ID

G=gap.SmallGroup(320,579);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,387,100,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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=b*e^9>;

// generators/relations

G = C24.9D10 order 320 = 26·5

9th non-split extension by C24 of D10 acting via D10/C5=C22

G = C₂⁴.₉D₁₀ order 320 = 2⁶·5

9^th non-split extension by C₂⁴ of D₁₀ acting via D₁₀/C₅=C₂²