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G = C10.1052- 1+4order 320 = 26·5

60th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.1052- 1+4, C20.429(C2×D4), (C2×C20).220D4, (C2×D4).235D10, (C2×Q8).193D10, Dic5⋊Q832C2, C20.17D430C2, C20.48D448C2, (C2×C20).650C23, (C2×C10).311C24, (C22×C4).285D10, C10.163(C22×D4), (C22×Dic10)⋊22C2, (D4×C10).314C22, C4⋊Dic5.320C22, (Q8×C10).240C22, C22.322(C23×D5), C23.208(C22×D5), C23.18D1032C2, C23.21D1035C2, (C22×C20).320C22, (C22×C10).237C23, C57(C23.38C23), (C2×Dic5).161C23, (C4×Dic5).182C22, C10.D4.92C22, C23.D5.133C22, C2.69(D4.10D10), (C2×Dic10).318C22, (C22×Dic5).166C22, C4.32(C2×C5⋊D4), (C2×C4○D4).11D5, (C2×C10).79(C2×D4), (C10×C4○D4).12C2, (C2×C4).97(C5⋊D4), C22.22(C2×C5⋊D4), C2.36(C22×C5⋊D4), (C2×C4).249(C22×D5), SmallGroup(320,1497)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.1052- 1+4
C1C5C10C2×C10C2×Dic5C22×Dic5C22×Dic10 — C10.1052- 1+4
C5C2×C10 — C10.1052- 1+4
C1C22C2×C4○D4

Generators and relations for C10.1052- 1+4
 G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=b2, e2=a5b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a5b-1, dbd-1=ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 782 in 270 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×10], C23, C23 [×2], C10, C10 [×2], C10 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic5 [×8], C20 [×4], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10, C22×C10 [×2], C23.38C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], C23.D5 [×10], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C20.48D4 [×4], C23.21D10, C23.18D10 [×4], C20.17D4 [×2], Dic5⋊Q8 [×2], C22×Dic10, C10×C4○D4, C10.1052- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], C5⋊D4 [×4], C22×D5 [×7], C23.38C23, C2×C5⋊D4 [×6], C23×D5, D4.10D10 [×2], C22×C5⋊D4, C10.1052- 1+4

Smallest permutation representation of C10.1052- 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)
(1 125 30 120)(2 124 21 119)(3 123 22 118)(4 122 23 117)(5 121 24 116)(6 130 25 115)(7 129 26 114)(8 128 27 113)(9 127 28 112)(10 126 29 111)(11 69 151 59)(12 68 152 58)(13 67 153 57)(14 66 154 56)(15 65 155 55)(16 64 156 54)(17 63 157 53)(18 62 158 52)(19 61 159 51)(20 70 160 60)(31 99 41 104)(32 98 42 103)(33 97 43 102)(34 96 44 101)(35 95 45 110)(36 94 46 109)(37 93 47 108)(38 92 48 107)(39 91 49 106)(40 100 50 105)(71 139 82 144)(72 138 83 143)(73 137 84 142)(74 136 85 141)(75 135 86 150)(76 134 87 149)(77 133 88 148)(78 132 89 147)(79 131 90 146)(80 140 81 145)
(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)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(51 104 56 109)(52 103 57 108)(53 102 58 107)(54 101 59 106)(55 110 60 105)(61 99 66 94)(62 98 67 93)(63 97 68 92)(64 96 69 91)(65 95 70 100)(71 124 76 129)(72 123 77 128)(73 122 78 127)(74 121 79 126)(75 130 80 125)(81 120 86 115)(82 119 87 114)(83 118 88 113)(84 117 89 112)(85 116 90 111)
(1 55 30 65)(2 56 21 66)(3 57 22 67)(4 58 23 68)(5 59 24 69)(6 60 25 70)(7 51 26 61)(8 52 27 62)(9 53 28 63)(10 54 29 64)(11 111 151 126)(12 112 152 127)(13 113 153 128)(14 114 154 129)(15 115 155 130)(16 116 156 121)(17 117 157 122)(18 118 158 123)(19 119 159 124)(20 120 160 125)(31 76 41 87)(32 77 42 88)(33 78 43 89)(34 79 44 90)(35 80 45 81)(36 71 46 82)(37 72 47 83)(38 73 48 84)(39 74 49 85)(40 75 50 86)(91 131 106 146)(92 132 107 147)(93 133 108 148)(94 134 109 149)(95 135 110 150)(96 136 101 141)(97 137 102 142)(98 138 103 143)(99 139 104 144)(100 140 105 145)
(1 125 25 115)(2 124 26 114)(3 123 27 113)(4 122 28 112)(5 121 29 111)(6 130 30 120)(7 129 21 119)(8 128 22 118)(9 127 23 117)(10 126 24 116)(11 54 156 69)(12 53 157 68)(13 52 158 67)(14 51 159 66)(15 60 160 65)(16 59 151 64)(17 58 152 63)(18 57 153 62)(19 56 154 61)(20 55 155 70)(31 99 46 109)(32 98 47 108)(33 97 48 107)(34 96 49 106)(35 95 50 105)(36 94 41 104)(37 93 42 103)(38 92 43 102)(39 91 44 101)(40 100 45 110)(71 144 87 134)(72 143 88 133)(73 142 89 132)(74 141 90 131)(75 150 81 140)(76 149 82 139)(77 148 83 138)(78 147 84 137)(79 146 85 136)(80 145 86 135)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4N5A5B10A···10F10G···10R20A···20H20I···20T
order122222224444444···45510···1010···1020···2020···20
size1111224422224420···20222···24···42···24···4

62 irreducible representations

dim1111111122222244
type+++++++++++++--
imageC1C2C2C2C2C2C2C2D4D5D10D10D10C5⋊D42- 1+4D4.10D10
kernelC10.1052- 1+4C20.48D4C23.21D10C23.18D10C20.17D4Dic5⋊Q8C22×Dic10C10×C4○D4C2×C20C2×C4○D4C22×C4C2×D4C2×Q8C2×C4C10C2
# reps14142211426621628

Matrix representation of C10.1052- 1+4 in GL6(𝔽41)

3100000
1740000
0025000
00292300
0000250
001402723
,
31400000
17100000
000871
00034435
007800
0032097
,
1010000
22310000
00191000
00132200
00031229
0019171919
,
100000
21400000
000010
0001036
0040000
00025040
,
1010000
22310000
000333440
0007376
007800
0032113234

G:=sub<GL(6,GF(41))| [31,17,0,0,0,0,0,4,0,0,0,0,0,0,25,29,0,14,0,0,0,23,0,0,0,0,0,0,25,27,0,0,0,0,0,23],[31,17,0,0,0,0,40,10,0,0,0,0,0,0,0,0,7,32,0,0,8,34,8,0,0,0,7,4,0,9,0,0,1,35,0,7],[10,22,0,0,0,0,1,31,0,0,0,0,0,0,19,13,0,19,0,0,10,22,31,17,0,0,0,0,22,19,0,0,0,0,9,19],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,25,0,0,1,0,0,0,0,0,0,36,0,40],[10,22,0,0,0,0,1,31,0,0,0,0,0,0,0,0,7,32,0,0,33,7,8,11,0,0,34,37,0,32,0,0,40,6,0,34] >;

C10.1052- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{105}2_-^{1+4}
% in TeX

G:=Group("C10.105ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1497);
// by ID

G=gap.SmallGroup(320,1497);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,570,12550]);
// Polycyclic

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