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G = C10.742- 1+4order 320 = 26·5

29th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.742- 1+4, C10.492+ 1+4, C4⋊D424D5, C4⋊C4.93D10, C202D429C2, (C2×D4).96D10, D10⋊Q816C2, (C2×C20).47C23, C22⋊C4.12D10, Dic5⋊D419C2, C20.48D445C2, C20.17D420C2, (C2×C10).165C24, (C22×C4).231D10, C4⋊Dic5.46C22, D10.12D423C2, C2.51(D46D10), Dic5.Q814C2, Dic5.5D423C2, (D4×C10).129C22, (C2×Dic5).82C23, (C22×D5).72C23, C23.115(C22×D5), C22.186(C23×D5), Dic5.14D421C2, C23.D5.29C22, D10⋊C4.18C22, C23.18D1023C2, C23.23D1013C2, (C22×C10).193C23, (C22×C20).313C22, C51(C22.56C24), (C4×Dic5).108C22, (C2×Dic10).36C22, C10.D4.79C22, C2.32(D4.10D10), (C22×Dic5).116C22, (C5×C4⋊D4)⋊27C2, (C2×C4×D5).99C22, (C2×C4).43(C22×D5), (C5×C4⋊C4).151C22, (C2×C5⋊D4).36C22, (C5×C22⋊C4).20C22, SmallGroup(320,1293)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.742- 1+4
C1C5C10C2×C10C22×D5C2×C4×D5D10.12D4 — C10.742- 1+4
C5C2×C10 — C10.742- 1+4
C1C22C4⋊D4

Generators and relations for C10.742- 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 790 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4 [×4], C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×3], C23, D5, C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×2], Dic5 [×7], C20 [×4], D10 [×3], C2×C10, C2×C10 [×9], C4⋊D4, C4⋊D4 [×3], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic10 [×2], C4×D5, C2×Dic5 [×7], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20 [×4], C2×C20, C5×D4 [×3], C22×D5, C22×C10 [×3], C22.56C24, C4×Dic5, C10.D4 [×7], C4⋊Dic5 [×2], D10⋊C4 [×3], C23.D5 [×7], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×3], C22×C20, D4×C10 [×3], Dic5.14D4 [×2], D10.12D4, Dic5.5D4, Dic5.Q8, D10⋊Q8, C20.48D4, C23.23D10, C23.18D10 [×2], C20.17D4, C202D4, Dic5⋊D4 [×2], C5×C4⋊D4, C10.742- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D5 [×7], C22.56C24, C23×D5, D46D10 [×2], D4.10D10, C10.742- 1+4

Smallest permutation representation of C10.742- 1+4
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 50 25 35)(2 49 26 34)(3 48 27 33)(4 47 28 32)(5 46 29 31)(6 45 30 40)(7 44 21 39)(8 43 22 38)(9 42 23 37)(10 41 24 36)(11 140 160 145)(12 139 151 144)(13 138 152 143)(14 137 153 142)(15 136 154 141)(16 135 155 150)(17 134 156 149)(18 133 157 148)(19 132 158 147)(20 131 159 146)(51 89 66 74)(52 88 67 73)(53 87 68 72)(54 86 69 71)(55 85 70 80)(56 84 61 79)(57 83 62 78)(58 82 63 77)(59 81 64 76)(60 90 65 75)(91 119 106 124)(92 118 107 123)(93 117 108 122)(94 116 109 121)(95 115 110 130)(96 114 101 129)(97 113 102 128)(98 112 103 127)(99 111 104 126)(100 120 105 125)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 141)(18 142)(19 143)(20 144)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 90 25 75)(2 89 26 74)(3 88 27 73)(4 87 28 72)(5 86 29 71)(6 85 30 80)(7 84 21 79)(8 83 22 78)(9 82 23 77)(10 81 24 76)(11 95 160 110)(12 94 151 109)(13 93 152 108)(14 92 153 107)(15 91 154 106)(16 100 155 105)(17 99 156 104)(18 98 157 103)(19 97 158 102)(20 96 159 101)(31 54 46 69)(32 53 47 68)(33 52 48 67)(34 51 49 66)(35 60 50 65)(36 59 41 64)(37 58 42 63)(38 57 43 62)(39 56 44 61)(40 55 45 70)(111 149 126 134)(112 148 127 133)(113 147 128 132)(114 146 129 131)(115 145 130 140)(116 144 121 139)(117 143 122 138)(118 142 123 137)(119 141 124 136)(120 150 125 135)
(1 11 25 160)(2 12 26 151)(3 13 27 152)(4 14 28 153)(5 15 29 154)(6 16 30 155)(7 17 21 156)(8 18 22 157)(9 19 23 158)(10 20 24 159)(31 146 46 131)(32 147 47 132)(33 148 48 133)(34 149 49 134)(35 150 50 135)(36 141 41 136)(37 142 42 137)(38 143 43 138)(39 144 44 139)(40 145 45 140)(51 116 66 121)(52 117 67 122)(53 118 68 123)(54 119 69 124)(55 120 70 125)(56 111 61 126)(57 112 62 127)(58 113 63 128)(59 114 64 129)(60 115 65 130)(71 96 86 101)(72 97 87 102)(73 98 88 103)(74 99 89 104)(75 100 90 105)(76 91 81 106)(77 92 82 107)(78 93 83 108)(79 94 84 109)(80 95 85 110)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4K5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order1222222244444···45510···10101010101010101020···2020202020
size111144420444420···20222···2444488884···48888

47 irreducible representations

dim1111111111111222224444
type+++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D5D10D10D10D102+ 1+42- 1+4D46D10D4.10D10
kernelC10.742- 1+4Dic5.14D4D10.12D4Dic5.5D4Dic5.Q8D10⋊Q8C20.48D4C23.23D10C23.18D10C20.17D4C202D4Dic5⋊D4C5×C4⋊D4C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C10C10C2C2
# reps1211111121121242262184

Matrix representation of C10.742- 1+4 in GL8(𝔽41)

400000000
040000000
004000000
000400000
00000600
000034700
00000006
000000347
,
040000000
10000000
000400000
00100000
000000740
000000734
000074000
000073400
,
01000000
10000000
00010000
00100000
00000010
00000001
00001000
00000100
,
000400000
00100000
040000000
10000000
00002294035
0000219261
0000161932
000015403922
,
00010000
00100000
040000000
400000000
000017600
0000342400
000000176
0000003424

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,7],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,22,2,1,15,0,0,0,0,9,19,6,40,0,0,0,0,40,26,19,39,0,0,0,0,35,1,32,22],[0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24] >;

C10.742- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{74}2_-^{1+4}
% in TeX

G:=Group("C10.74ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1293);
// by ID

G=gap.SmallGroup(320,1293);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^5*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations

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