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G = C10.1062- 1+4order 320 = 26·5

61st non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.1062- 1+4, C10.1442+ 1+4, C4○D45Dic5, D48(C2×Dic5), Q87(C2×Dic5), (Q8×Dic5)⋊28C2, (D4×Dic5)⋊41C2, (C2×D4).253D10, C10.72(C23×C4), (C2×Q8).209D10, C2.5(D48D10), (C2×C10).313C24, (C2×C20).560C23, C20.159(C22×C4), (C22×C4).286D10, C4.22(C22×Dic5), C2.13(C23×Dic5), C22.49(C23×D5), (D4×C10).275C22, C4⋊Dic5.392C22, (Q8×C10).242C22, C23.210(C22×D5), C2.5(D4.10D10), C23.21D1037C2, C22.4(C22×Dic5), (C22×C20).295C22, (C22×C10).239C23, C56(C23.33C23), (C4×Dic5).183C22, (C2×Dic5).302C23, C23.D5.135C22, (C22×Dic5).167C22, (C2×C20)⋊30(C2×C4), (C5×C4○D4)⋊12C4, (C5×D4)⋊32(C2×C4), (C5×Q8)⋊29(C2×C4), (C2×C4)⋊5(C2×Dic5), (C2×C4⋊Dic5)⋊47C2, (C2×C4○D4).12D5, (C10×C4○D4).14C2, (C2×C4).638(C22×D5), (C2×C10).132(C22×C4), SmallGroup(320,1499)

Series: Derived Chief Lower central Upper central

C1C10 — C10.1062- 1+4
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — C10.1062- 1+4
C5C10 — C10.1062- 1+4
C1C22C2×C4○D4

Generators and relations for C10.1062- 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 734 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C10 [×3], C10 [×6], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×8], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic5 [×8], C2×Dic5 [×6], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C23.33C23, C4×Dic5 [×6], C4⋊Dic5, C4⋊Dic5 [×9], C23.D5 [×6], C22×Dic5 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4⋊Dic5 [×3], C23.21D10 [×3], D4×Dic5 [×6], Q8×Dic5 [×2], C10×C4○D4, C10.1062- 1+4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C2×Dic5 [×28], C22×D5 [×7], C23.33C23, C22×Dic5 [×14], C23×D5, D48D10, D4.10D10, C23×Dic5, C10.1062- 1+4

Smallest permutation representation of C10.1062- 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 78 30 81)(2 79 21 82)(3 80 22 83)(4 71 23 84)(5 72 24 85)(6 73 25 86)(7 74 26 87)(8 75 27 88)(9 76 28 89)(10 77 29 90)(11 106 151 91)(12 107 152 92)(13 108 153 93)(14 109 154 94)(15 110 155 95)(16 101 156 96)(17 102 157 97)(18 103 158 98)(19 104 159 99)(20 105 160 100)(31 63 43 51)(32 64 44 52)(33 65 45 53)(34 66 46 54)(35 67 47 55)(36 68 48 56)(37 69 49 57)(38 70 50 58)(39 61 41 59)(40 62 42 60)(111 131 126 146)(112 132 127 147)(113 133 128 148)(114 134 129 149)(115 135 130 150)(116 136 121 141)(117 137 122 142)(118 138 123 143)(119 139 124 144)(120 140 125 145)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 71)(10 72)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 68)(32 69)(33 70)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(1 50 30 38)(2 41 21 39)(3 42 22 40)(4 43 23 31)(5 44 24 32)(6 45 25 33)(7 46 26 34)(8 47 27 35)(9 48 28 36)(10 49 29 37)(11 131 151 146)(12 132 152 147)(13 133 153 148)(14 134 154 149)(15 135 155 150)(16 136 156 141)(17 137 157 142)(18 138 158 143)(19 139 159 144)(20 140 160 145)(51 84 63 71)(52 85 64 72)(53 86 65 73)(54 87 66 74)(55 88 67 75)(56 89 68 76)(57 90 69 77)(58 81 70 78)(59 82 61 79)(60 83 62 80)(91 111 106 126)(92 112 107 127)(93 113 108 128)(94 114 109 129)(95 115 110 130)(96 116 101 121)(97 117 102 122)(98 118 103 123)(99 119 104 124)(100 120 105 125)
(1 140 6 135)(2 139 7 134)(3 138 8 133)(4 137 9 132)(5 136 10 131)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 144 26 149)(22 143 27 148)(23 142 28 147)(24 141 29 146)(25 150 30 145)(31 157 36 152)(32 156 37 151)(33 155 38 160)(34 154 39 159)(35 153 40 158)(51 102 56 107)(52 101 57 106)(53 110 58 105)(54 109 59 104)(55 108 60 103)(61 99 66 94)(62 98 67 93)(63 97 68 92)(64 96 69 91)(65 95 70 100)(71 122 76 127)(72 121 77 126)(73 130 78 125)(74 129 79 124)(75 128 80 123)(81 120 86 115)(82 119 87 114)(83 118 88 113)(84 117 89 112)(85 116 90 111)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4X5A5B10A···10F10G···10R20A···20H20I···20T
order12222···24···44···45510···1010···1020···2020···20
size11112···22···210···10222···24···42···24···4

74 irreducible representations

dim1111111222224444
type++++++++++-+-+-
imageC1C2C2C2C2C2C4D5D10D10D10Dic52+ 1+42- 1+4D48D10D4.10D10
kernelC10.1062- 1+4C2×C4⋊Dic5C23.21D10D4×Dic5Q8×Dic5C10×C4○D4C5×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C10C10C2C2
# reps133621162662161144

Matrix representation of C10.1062- 1+4 in GL6(𝔽41)

0340000
6350000
0040700
0034700
0061367
00822350
,
100000
010000
00951821
001138021
001227234
0032341433
,
100000
010000
0032362320
00303020
00385397
0093278
,
4000000
0400000
00113200
0093000
00436232
001913739
,
2310000
3180000
00141400
00302700
0011262514
0022331416

G:=sub<GL(6,GF(41))| [0,6,0,0,0,0,34,35,0,0,0,0,0,0,40,34,6,8,0,0,7,7,13,22,0,0,0,0,6,35,0,0,0,0,7,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,12,32,0,0,5,38,27,34,0,0,18,0,2,14,0,0,21,21,34,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,30,38,9,0,0,36,3,5,3,0,0,23,0,39,27,0,0,20,20,7,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,9,4,19,0,0,32,30,36,1,0,0,0,0,2,37,0,0,0,0,32,39],[23,3,0,0,0,0,1,18,0,0,0,0,0,0,14,30,11,22,0,0,14,27,26,33,0,0,0,0,25,14,0,0,0,0,14,16] >;

C10.1062- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{106}2_-^{1+4}
% in TeX

G:=Group("C10.106ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1499);
// by ID

G=gap.SmallGroup(320,1499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,12550]);
// Polycyclic

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

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