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## G = C10.432+ 1+4order 320 = 26·5

### 43rd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.432+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C10.432+ 1+4
 Lower central C5 — C2×C10 — C10.432+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.432+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, bd=db, ebe-1=a5b, cd=dc, ece-1=a5c, ede-1=b2d >

Subgroups: 790 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×10], C23, C23 [×2], C23, D5 [×2], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], Dic5 [×7], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×9], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×C10, C22×C10 [×2], C22.47C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4, C23.D5, C23.D5 [×6], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.D10 [×2], Dic54D4 [×2], C4.Dic10, D5×C4⋊C4, C23.21D10, C4×C5⋊D4, D4×Dic5, C23.18D10 [×2], C202D4, C202D4 [×2], C5×C4⋊D4, C10.432+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, D42D5 [×2], C23×D5, C2×D42D5, D46D10, D5×C4○D4, C10.432+ 1+4

Smallest permutation representation of C10.432+ 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 16 25 155)(2 15 26 154)(3 14 27 153)(4 13 28 152)(5 12 29 151)(6 11 30 160)(7 20 21 159)(8 19 22 158)(9 18 23 157)(10 17 24 156)(31 134 46 149)(32 133 47 148)(33 132 48 147)(34 131 49 146)(35 140 50 145)(36 139 41 144)(37 138 42 143)(38 137 43 142)(39 136 44 141)(40 135 45 150)(51 129 66 114)(52 128 67 113)(53 127 68 112)(54 126 69 111)(55 125 70 120)(56 124 61 119)(57 123 62 118)(58 122 63 117)(59 121 64 116)(60 130 65 115)(71 94 86 109)(72 93 87 108)(73 92 88 107)(74 91 89 106)(75 100 90 105)(76 99 81 104)(77 98 82 103)(78 97 83 102)(79 96 84 101)(80 95 85 110)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 150)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(61 86)(62 87)(63 88)(64 89)(65 90)(66 81)(67 82)(68 83)(69 84)(70 85)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)
(1 130 30 120)(2 121 21 111)(3 122 22 112)(4 123 23 113)(5 124 24 114)(6 125 25 115)(7 126 26 116)(8 127 27 117)(9 128 28 118)(10 129 29 119)(11 70 155 60)(12 61 156 51)(13 62 157 52)(14 63 158 53)(15 64 159 54)(16 65 160 55)(17 66 151 56)(18 67 152 57)(19 68 153 58)(20 69 154 59)(31 101 41 91)(32 102 42 92)(33 103 43 93)(34 104 44 94)(35 105 45 95)(36 106 46 96)(37 107 47 97)(38 108 48 98)(39 109 49 99)(40 110 50 100)(71 146 81 136)(72 147 82 137)(73 148 83 138)(74 149 84 139)(75 150 85 140)(76 141 86 131)(77 142 87 132)(78 143 88 133)(79 144 89 134)(80 145 90 135)
(1 145 6 150)(2 146 7 141)(3 147 8 142)(4 148 9 143)(5 149 10 144)(11 35 16 40)(12 36 17 31)(13 37 18 32)(14 38 19 33)(15 39 20 34)(21 136 26 131)(22 137 27 132)(23 138 28 133)(24 139 29 134)(25 140 30 135)(41 156 46 151)(42 157 47 152)(43 158 48 153)(44 159 49 154)(45 160 50 155)(51 101 56 106)(52 102 57 107)(53 103 58 108)(54 104 59 109)(55 105 60 110)(61 91 66 96)(62 92 67 97)(63 93 68 98)(64 94 69 99)(65 95 70 100)(71 121 76 126)(72 122 77 127)(73 123 78 128)(74 124 79 129)(75 125 80 130)(81 111 86 116)(82 112 87 117)(83 113 88 118)(84 114 89 119)(85 115 90 120)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 20A ··· 20H 20I 20J 20K 20L order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 4 10 10 2 2 2 2 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 D5×C4○D4 kernel C10.432+ 1+4 C23.D10 Dic5⋊4D4 C4.Dic10 D5×C4⋊C4 C23.21D10 C4×C5⋊D4 D4×Dic5 C23.18D10 C20⋊2D4 C5×C4⋊D4 C4⋊D4 C20 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C2 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 2 4 4 4 2 2 6 1 4 4 4

Matrix representation of C10.432+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 0 0 0 0 7 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 7 40 0 0 0 0 7 34 0 0 0 0 0 0 40 0 0 0 0 0 40 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 39 0 0 0 0 0 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 39 0 0 0 0 1 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,0,0,0,0,40,40,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,39,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40] >;`

C10.432+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{43}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.43ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1286);`
`// by ID`

`G=gap.SmallGroup(320,1286);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,185,12550]);`
`// Polycyclic`

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

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