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G = C24.21D10  order 320 = 26·5

21st non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.21D10, (C2×C20)⋊23D4, (C2×Dic5)⋊7D4, (C22×D5)⋊5D4, (C22×D4)⋊2D5, (C22×C10)⋊8D4, C23⋊3(C5⋊D4), C5⋊3(C23⋊2D4), C10.68C22≀C2, C22.284(D4×D5), C2.26(C20⋊2D4), C2.27(C20⋊D4), C10.37(C4⋊1D4), (C22×C4).153D10, C2.7(C24⋊2D5), C2.35(C23⋊D10), C10.131(C4⋊D4), (C23×C10).49C22, (C23×D5).25C22, C23.385(C22×D5), C10.10C42⋊46C2, C2.35(Dic5⋊D4), (C22×C20).396C22, (C22×C10).368C23, C22.107(D4⋊2D5), (C22×Dic5).69C22, (D4×C2×C10)⋊12C2, (C2×C4)⋊5(C5⋊D4), (C22×C5⋊D4)⋊2C2, (C2×C10).557(C2×D4), (C2×C23.D5)⋊12C2, (C2×D10⋊C4)⋊39C2, C22.219(C2×C5⋊D4), (C2×C10).164(C4○D4), SmallGroup(320,850)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — C24.21D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×C5⋊D4 — C24.21D10
Lower central
C5 — C22×C10 — C24.21D10
Upper central
C1 — C23 — C22×D4

Generators and relations for C24.21D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 1270 in 322 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C22×D4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23⋊2D4, D10⋊C4, C23.D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23×C10, C10.10C42, C2×D10⋊C4, C2×C23.D5, C22×C5⋊D4, D4×C2×C10, C24.21D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C4⋊1D4, C5⋊D4, C22×D5, C23⋊2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C23⋊D10, C20⋊2D4, Dic5⋊D4, C20⋊D4, C24⋊2D5, C24.21D10

Smallest permutation representation of C24.21D10
►On 160 points
Generators in S160
(1 38)(2 144)(3 40)(4 146)(5 32)(6 148)(7 34)(8 150)(9 36)(10 142)(11 67)(12 109)(13 69)(14 101)(15 61)(16 103)(17 63)(18 105)(19 65)(20 107)(21 156)(22 42)(23 158)(24 44)(25 160)(26 46)(27 152)(28 48)(29 154)(30 50)(31 126)(33 128)(35 130)(37 122)(39 124)(41 136)(43 138)(45 140)(47 132)(49 134)(51 112)(52 72)(53 114)(54 74)(55 116)(56 76)(57 118)(58 78)(59 120)(60 80)(62 88)(64 90)(66 82)(68 84)(70 86)(71 92)(73 94)(75 96)(77 98)(79 100)(81 106)(83 108)(85 110)(87 102)(89 104)(91 111)(93 113)(95 115)(97 117)(99 119)(121 141)(123 143)(125 145)(127 147)(129 149)(131 151)(133 153)(135 155)(137 157)(139 159)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C···4H5A5B10A···10N10O···10AD20A···20H
order12···2222222444···45510···1010···1020···20
size11···1444420204420···20222···24···44···4

62 irreducible representations

dim111111222222222244
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D4D5C4○D4D10D10C5⋊D4C5⋊D4D4×D5D4⋊2D5
kernelC24.21D10C10.10C42C2×D10⋊C4C2×C23.D5C22×C5⋊D4D4×C2×C10C2×Dic5C2×C20C22×D5C22×C10C22×D4C2×C10C22×C4C24C2×C4C23C22C22
# reps1112214224222481662

Matrix representation of C24.21D10 ►in GL6(𝔽41)

100000
010000
0017100
00402400
000010
00003640
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
001000
000100
0000400
0000040
,
24370000
31170000
0013400
0073400
0000920
00003732
,
1740000
30240000
001000
0074000
00003221
000009

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,1,36,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,31,0,0,0,0,37,17,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,9,37,0,0,0,0,20,32],[17,30,0,0,0,0,4,24,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,21,9] >;
 

C24.21D10 in GAP, Magma, Sage, TeX 

C_2^4._{21}D_{10}
 
% in TeX
 
G:=Group("C2^4.21D10");
 
// GroupNames label
 
G:=SmallGroup(320,850);
 
// by ID
 
G=gap.SmallGroup(320,850);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,387,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*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=c*e^-1>;
 
// generators/relations
 

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