G = C10.1052- 1+4 order 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⋊Q8⋊32C2, C20.17D4⋊30C2, C20.48D4⋊48C2, (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), C23.18D10⋊32C2, C23.21D10⋊35C2, (C22×C20).320C22, (C22×C10).237C23, C5⋊7(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
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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C23.38C23, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C20.48D4, C23.21D10, C23.18D10, C20.17D4, Dic5⋊Q8, C22×Dic10, C10×C4○D4, C10.1052- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2- 1+4, C5⋊D4, C22×D5, C23.38C23, C2×C5⋊D4, C23×D5, D4.10D10, C22×C5⋊D4, 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 30 118)(2 124 21 117)(3 123 22 116)(4 122 23 115)(5 121 24 114)(6 130 25 113)(7 129 26 112)(8 128 27 111)(9 127 28 120)(10 126 29 119)(11 69 159 51)(12 68 160 60)(13 67 151 59)(14 66 152 58)(15 65 153 57)(16 64 154 56)(17 63 155 55)(18 62 156 54)(19 61 157 53)(20 70 158 52)(31 99 49 106)(32 98 50 105)(33 97 41 104)(34 96 42 103)(35 95 43 102)(36 94 44 101)(37 93 45 110)(38 92 46 109)(39 91 47 108)(40 100 48 107)(71 139 90 146)(72 138 81 145)(73 137 82 144)(74 136 83 143)(75 135 84 142)(76 134 85 141)(77 133 86 150)(78 132 87 149)(79 131 88 148)(80 140 89 147)
(1 138 6 133)(2 137 7 132)(3 136 8 131)(4 135 9 140)(5 134 10 139)(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)(51 106 56 101)(52 105 57 110)(53 104 58 109)(54 103 59 108)(55 102 60 107)(61 97 66 92)(62 96 67 91)(63 95 68 100)(64 94 69 99)(65 93 70 98)(71 126 76 121)(72 125 77 130)(73 124 78 129)(74 123 79 128)(75 122 80 127)(81 118 86 113)(82 117 87 112)(83 116 88 111)(84 115 89 120)(85 114 90 119)
(1 57 30 65)(2 58 21 66)(3 59 22 67)(4 60 23 68)(5 51 24 69)(6 52 25 70)(7 53 26 61)(8 54 27 62)(9 55 28 63)(10 56 29 64)(11 119 159 126)(12 120 160 127)(13 111 151 128)(14 112 152 129)(15 113 153 130)(16 114 154 121)(17 115 155 122)(18 116 156 123)(19 117 157 124)(20 118 158 125)(31 76 49 85)(32 77 50 86)(33 78 41 87)(34 79 42 88)(35 80 43 89)(36 71 44 90)(37 72 45 81)(38 73 46 82)(39 74 47 83)(40 75 48 84)(91 131 108 148)(92 132 109 149)(93 133 110 150)(94 134 101 141)(95 135 102 142)(96 136 103 143)(97 137 104 144)(98 138 105 145)(99 139 106 146)(100 140 107 147)
(1 125 25 113)(2 124 26 112)(3 123 27 111)(4 122 28 120)(5 121 29 119)(6 130 30 118)(7 129 21 117)(8 128 22 116)(9 127 23 115)(10 126 24 114)(11 56 154 69)(12 55 155 68)(13 54 156 67)(14 53 157 66)(15 52 158 65)(16 51 159 64)(17 60 160 63)(18 59 151 62)(19 58 152 61)(20 57 153 70)(31 99 44 101)(32 98 45 110)(33 97 46 109)(34 96 47 108)(35 95 48 107)(36 94 49 106)(37 93 50 105)(38 92 41 104)(39 91 42 103)(40 100 43 102)(71 146 85 134)(72 145 86 133)(73 144 87 132)(74 143 88 131)(75 142 89 140)(76 141 90 139)(77 150 81 138)(78 149 82 137)(79 148 83 136)(80 147 84 135)
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), (1,125,30,118)(2,124,21,117)(3,123,22,116)(4,122,23,115)(5,121,24,114)(6,130,25,113)(7,129,26,112)(8,128,27,111)(9,127,28,120)(10,126,29,119)(11,69,159,51)(12,68,160,60)(13,67,151,59)(14,66,152,58)(15,65,153,57)(16,64,154,56)(17,63,155,55)(18,62,156,54)(19,61,157,53)(20,70,158,52)(31,99,49,106)(32,98,50,105)(33,97,41,104)(34,96,42,103)(35,95,43,102)(36,94,44,101)(37,93,45,110)(38,92,46,109)(39,91,47,108)(40,100,48,107)(71,139,90,146)(72,138,81,145)(73,137,82,144)(74,136,83,143)(75,135,84,142)(76,134,85,141)(77,133,86,150)(78,132,87,149)(79,131,88,148)(80,140,89,147), (1,138,6,133)(2,137,7,132)(3,136,8,131)(4,135,9,140)(5,134,10,139)(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)(51,106,56,101)(52,105,57,110)(53,104,58,109)(54,103,59,108)(55,102,60,107)(61,97,66,92)(62,96,67,91)(63,95,68,100)(64,94,69,99)(65,93,70,98)(71,126,76,121)(72,125,77,130)(73,124,78,129)(74,123,79,128)(75,122,80,127)(81,118,86,113)(82,117,87,112)(83,116,88,111)(84,115,89,120)(85,114,90,119), (1,57,30,65)(2,58,21,66)(3,59,22,67)(4,60,23,68)(5,51,24,69)(6,52,25,70)(7,53,26,61)(8,54,27,62)(9,55,28,63)(10,56,29,64)(11,119,159,126)(12,120,160,127)(13,111,151,128)(14,112,152,129)(15,113,153,130)(16,114,154,121)(17,115,155,122)(18,116,156,123)(19,117,157,124)(20,118,158,125)(31,76,49,85)(32,77,50,86)(33,78,41,87)(34,79,42,88)(35,80,43,89)(36,71,44,90)(37,72,45,81)(38,73,46,82)(39,74,47,83)(40,75,48,84)(91,131,108,148)(92,132,109,149)(93,133,110,150)(94,134,101,141)(95,135,102,142)(96,136,103,143)(97,137,104,144)(98,138,105,145)(99,139,106,146)(100,140,107,147), (1,125,25,113)(2,124,26,112)(3,123,27,111)(4,122,28,120)(5,121,29,119)(6,130,30,118)(7,129,21,117)(8,128,22,116)(9,127,23,115)(10,126,24,114)(11,56,154,69)(12,55,155,68)(13,54,156,67)(14,53,157,66)(15,52,158,65)(16,51,159,64)(17,60,160,63)(18,59,151,62)(19,58,152,61)(20,57,153,70)(31,99,44,101)(32,98,45,110)(33,97,46,109)(34,96,47,108)(35,95,48,107)(36,94,49,106)(37,93,50,105)(38,92,41,104)(39,91,42,103)(40,100,43,102)(71,146,85,134)(72,145,86,133)(73,144,87,132)(74,143,88,131)(75,142,89,140)(76,141,90,139)(77,150,81,138)(78,149,82,137)(79,148,83,136)(80,147,84,135)>;
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), (1,125,30,118)(2,124,21,117)(3,123,22,116)(4,122,23,115)(5,121,24,114)(6,130,25,113)(7,129,26,112)(8,128,27,111)(9,127,28,120)(10,126,29,119)(11,69,159,51)(12,68,160,60)(13,67,151,59)(14,66,152,58)(15,65,153,57)(16,64,154,56)(17,63,155,55)(18,62,156,54)(19,61,157,53)(20,70,158,52)(31,99,49,106)(32,98,50,105)(33,97,41,104)(34,96,42,103)(35,95,43,102)(36,94,44,101)(37,93,45,110)(38,92,46,109)(39,91,47,108)(40,100,48,107)(71,139,90,146)(72,138,81,145)(73,137,82,144)(74,136,83,143)(75,135,84,142)(76,134,85,141)(77,133,86,150)(78,132,87,149)(79,131,88,148)(80,140,89,147), (1,138,6,133)(2,137,7,132)(3,136,8,131)(4,135,9,140)(5,134,10,139)(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)(51,106,56,101)(52,105,57,110)(53,104,58,109)(54,103,59,108)(55,102,60,107)(61,97,66,92)(62,96,67,91)(63,95,68,100)(64,94,69,99)(65,93,70,98)(71,126,76,121)(72,125,77,130)(73,124,78,129)(74,123,79,128)(75,122,80,127)(81,118,86,113)(82,117,87,112)(83,116,88,111)(84,115,89,120)(85,114,90,119), (1,57,30,65)(2,58,21,66)(3,59,22,67)(4,60,23,68)(5,51,24,69)(6,52,25,70)(7,53,26,61)(8,54,27,62)(9,55,28,63)(10,56,29,64)(11,119,159,126)(12,120,160,127)(13,111,151,128)(14,112,152,129)(15,113,153,130)(16,114,154,121)(17,115,155,122)(18,116,156,123)(19,117,157,124)(20,118,158,125)(31,76,49,85)(32,77,50,86)(33,78,41,87)(34,79,42,88)(35,80,43,89)(36,71,44,90)(37,72,45,81)(38,73,46,82)(39,74,47,83)(40,75,48,84)(91,131,108,148)(92,132,109,149)(93,133,110,150)(94,134,101,141)(95,135,102,142)(96,136,103,143)(97,137,104,144)(98,138,105,145)(99,139,106,146)(100,140,107,147), (1,125,25,113)(2,124,26,112)(3,123,27,111)(4,122,28,120)(5,121,29,119)(6,130,30,118)(7,129,21,117)(8,128,22,116)(9,127,23,115)(10,126,24,114)(11,56,154,69)(12,55,155,68)(13,54,156,67)(14,53,157,66)(15,52,158,65)(16,51,159,64)(17,60,160,63)(18,59,151,62)(19,58,152,61)(20,57,153,70)(31,99,44,101)(32,98,45,110)(33,97,46,109)(34,96,47,108)(35,95,48,107)(36,94,49,106)(37,93,50,105)(38,92,41,104)(39,91,42,103)(40,100,43,102)(71,146,85,134)(72,145,86,133)(73,144,87,132)(74,143,88,131)(75,142,89,140)(76,141,90,139)(77,150,81,138)(78,149,82,137)(79,148,83,136)(80,147,84,135) );
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)], [(1,125,30,118),(2,124,21,117),(3,123,22,116),(4,122,23,115),(5,121,24,114),(6,130,25,113),(7,129,26,112),(8,128,27,111),(9,127,28,120),(10,126,29,119),(11,69,159,51),(12,68,160,60),(13,67,151,59),(14,66,152,58),(15,65,153,57),(16,64,154,56),(17,63,155,55),(18,62,156,54),(19,61,157,53),(20,70,158,52),(31,99,49,106),(32,98,50,105),(33,97,41,104),(34,96,42,103),(35,95,43,102),(36,94,44,101),(37,93,45,110),(38,92,46,109),(39,91,47,108),(40,100,48,107),(71,139,90,146),(72,138,81,145),(73,137,82,144),(74,136,83,143),(75,135,84,142),(76,134,85,141),(77,133,86,150),(78,132,87,149),(79,131,88,148),(80,140,89,147)], [(1,138,6,133),(2,137,7,132),(3,136,8,131),(4,135,9,140),(5,134,10,139),(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),(51,106,56,101),(52,105,57,110),(53,104,58,109),(54,103,59,108),(55,102,60,107),(61,97,66,92),(62,96,67,91),(63,95,68,100),(64,94,69,99),(65,93,70,98),(71,126,76,121),(72,125,77,130),(73,124,78,129),(74,123,79,128),(75,122,80,127),(81,118,86,113),(82,117,87,112),(83,116,88,111),(84,115,89,120),(85,114,90,119)], [(1,57,30,65),(2,58,21,66),(3,59,22,67),(4,60,23,68),(5,51,24,69),(6,52,25,70),(7,53,26,61),(8,54,27,62),(9,55,28,63),(10,56,29,64),(11,119,159,126),(12,120,160,127),(13,111,151,128),(14,112,152,129),(15,113,153,130),(16,114,154,121),(17,115,155,122),(18,116,156,123),(19,117,157,124),(20,118,158,125),(31,76,49,85),(32,77,50,86),(33,78,41,87),(34,79,42,88),(35,80,43,89),(36,71,44,90),(37,72,45,81),(38,73,46,82),(39,74,47,83),(40,75,48,84),(91,131,108,148),(92,132,109,149),(93,133,110,150),(94,134,101,141),(95,135,102,142),(96,136,103,143),(97,137,104,144),(98,138,105,145),(99,139,106,146),(100,140,107,147)], [(1,125,25,113),(2,124,26,112),(3,123,27,111),(4,122,28,120),(5,121,29,119),(6,130,30,118),(7,129,21,117),(8,128,22,116),(9,127,23,115),(10,126,24,114),(11,56,154,69),(12,55,155,68),(13,54,156,67),(14,53,157,66),(15,52,158,65),(16,51,159,64),(17,60,160,63),(18,59,151,62),(19,58,152,61),(20,57,153,70),(31,99,44,101),(32,98,45,110),(33,97,46,109),(34,96,47,108),(35,95,48,107),(36,94,49,106),(37,93,50,105),(38,92,41,104),(39,91,42,103),(40,100,43,102),(71,146,85,134),(72,145,86,133),(73,144,87,132),(74,143,88,131),(75,142,89,140),(76,141,90,139),(77,150,81,138),(78,149,82,137),(79,148,83,136),(80,147,84,135)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | 2- 1+4 | D4.10D10 |
| kernel | C10.1052- 1+4 | C20.48D4 | C23.21D10 | C23.18D10 | C20.17D4 | Dic5⋊Q8 | C22×Dic10 | C10×C4○D4 | C2×C20 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C10 | C2 |
| # reps | 1 | 4 | 1 | 4 | 2 | 2 | 1 | 1 | 4 | 2 | 6 | 6 | 2 | 16 | 2 | 8 |
Matrix representation of C10.1052- 1+4 ►in GL6(𝔽41)
| 31 | 0 | 0 | 0 | 0 | 0 |
| 17 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 29 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 14 | 0 | 27 | 23 |
| 31 | 40 | 0 | 0 | 0 | 0 |
| 17 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 7 | 1 |
| 0 | 0 | 0 | 34 | 4 | 35 |
| 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 32 | 0 | 9 | 7 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 22 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 13 | 22 | 0 | 0 |
| 0 | 0 | 0 | 31 | 22 | 9 |
| 0 | 0 | 19 | 17 | 19 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 36 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 40 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 22 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 33 | 34 | 40 |
| 0 | 0 | 0 | 7 | 37 | 6 |
| 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 32 | 11 | 32 | 34 |
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