G = C10.112+ 1+4 order 320 = 26·5
11st non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.112+ 1+4, C4⋊D20⋊10C2, C20⋊7D4⋊28C2, C4⋊C4.307D10, D20⋊8C4⋊10C2, C4.94(C4○D20), (C2×C10).58C24, C4.Dic10⋊10C2, D10.13D4⋊2C2, C20.196(C4○D4), (C2×C20).619C23, (C22×C4).183D10, C2.14(D4⋊6D10), C22.92(C23×D5), (C2×D20).142C22, C4⋊Dic5.194C22, (C4×Dic5).74C22, (C2×Dic5).19C23, (C22×D5).16C23, C23.229(C22×D5), (C22×C10).407C23, (C22×C20).220C22, C5⋊1(C22.47C24), C22.11(Q8⋊2D5), C23.D5.143C22, D10⋊C4.142C22, C10.D4.151C22, (C2×C4⋊C4)⋊23D5, (C10×C4⋊C4)⋊20C2, C4⋊C4⋊D5⋊2C2, (C4×C5⋊D4)⋊11C2, C4⋊C4⋊7D5⋊10C2, C2.27(C2×C4○D20), C10.25(C2×C4○D4), C2.9(C2×Q8⋊2D5), (C2×C4×D5).66C22, (C5×C4⋊C4).299C22, (C2×C4).146(C22×D5), (C2×C10).198(C4○D4), (C2×C5⋊D4).103C22, SmallGroup(320,1186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.112+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 870 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.47C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C4.Dic10, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4, C4⋊D20, C4⋊C4⋊D5, C4×C5⋊D4, C4×C5⋊D4, C20⋊7D4, C20⋊7D4, C10×C4⋊C4, C10.112+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, Q8⋊2D5, C23×D5, C2×C4○D20, D4⋊6D10, C2×Q8⋊2D5, C10.112+ 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 90 30 80)(2 89 21 79)(3 88 22 78)(4 87 23 77)(5 86 24 76)(6 85 25 75)(7 84 26 74)(8 83 27 73)(9 82 28 72)(10 81 29 71)(11 105 155 95)(12 104 156 94)(13 103 157 93)(14 102 158 92)(15 101 159 91)(16 110 160 100)(17 109 151 99)(18 108 152 98)(19 107 153 97)(20 106 154 96)(31 54 41 64)(32 53 42 63)(33 52 43 62)(34 51 44 61)(35 60 45 70)(36 59 46 69)(37 58 47 68)(38 57 48 67)(39 56 49 66)(40 55 50 65)(111 134 121 144)(112 133 122 143)(113 132 123 142)(114 131 124 141)(115 140 125 150)(116 139 126 149)(117 138 127 148)(118 137 128 147)(119 136 129 146)(120 135 130 145)
(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 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(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)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 100 30 110)(2 91 21 101)(3 92 22 102)(4 93 23 103)(5 94 24 104)(6 95 25 105)(7 96 26 106)(8 97 27 107)(9 98 28 108)(10 99 29 109)(11 75 155 85)(12 76 156 86)(13 77 157 87)(14 78 158 88)(15 79 159 89)(16 80 160 90)(17 71 151 81)(18 72 152 82)(19 73 153 83)(20 74 154 84)(31 111 41 121)(32 112 42 122)(33 113 43 123)(34 114 44 124)(35 115 45 125)(36 116 46 126)(37 117 47 127)(38 118 48 128)(39 119 49 129)(40 120 50 130)(51 131 61 141)(52 132 62 142)(53 133 63 143)(54 134 64 144)(55 135 65 145)(56 136 66 146)(57 137 67 147)(58 138 68 148)(59 139 69 149)(60 140 70 150)
(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), (1,90,30,80)(2,89,21,79)(3,88,22,78)(4,87,23,77)(5,86,24,76)(6,85,25,75)(7,84,26,74)(8,83,27,73)(9,82,28,72)(10,81,29,71)(11,105,155,95)(12,104,156,94)(13,103,157,93)(14,102,158,92)(15,101,159,91)(16,110,160,100)(17,109,151,99)(18,108,152,98)(19,107,153,97)(20,106,154,96)(31,54,41,64)(32,53,42,63)(33,52,43,62)(34,51,44,61)(35,60,45,70)(36,59,46,69)(37,58,47,68)(38,57,48,67)(39,56,49,66)(40,55,50,65)(111,134,121,144)(112,133,122,143)(113,132,123,142)(114,131,124,141)(115,140,125,150)(116,139,126,149)(117,138,127,148)(118,137,128,147)(119,136,129,146)(120,135,130,145), (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)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(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)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,100,30,110)(2,91,21,101)(3,92,22,102)(4,93,23,103)(5,94,24,104)(6,95,25,105)(7,96,26,106)(8,97,27,107)(9,98,28,108)(10,99,29,109)(11,75,155,85)(12,76,156,86)(13,77,157,87)(14,78,158,88)(15,79,159,89)(16,80,160,90)(17,71,151,81)(18,72,152,82)(19,73,153,83)(20,74,154,84)(31,111,41,121)(32,112,42,122)(33,113,43,123)(34,114,44,124)(35,115,45,125)(36,116,46,126)(37,117,47,127)(38,118,48,128)(39,119,49,129)(40,120,50,130)(51,131,61,141)(52,132,62,142)(53,133,63,143)(54,134,64,144)(55,135,65,145)(56,136,66,146)(57,137,67,147)(58,138,68,148)(59,139,69,149)(60,140,70,150), (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), (1,90,30,80)(2,89,21,79)(3,88,22,78)(4,87,23,77)(5,86,24,76)(6,85,25,75)(7,84,26,74)(8,83,27,73)(9,82,28,72)(10,81,29,71)(11,105,155,95)(12,104,156,94)(13,103,157,93)(14,102,158,92)(15,101,159,91)(16,110,160,100)(17,109,151,99)(18,108,152,98)(19,107,153,97)(20,106,154,96)(31,54,41,64)(32,53,42,63)(33,52,43,62)(34,51,44,61)(35,60,45,70)(36,59,46,69)(37,58,47,68)(38,57,48,67)(39,56,49,66)(40,55,50,65)(111,134,121,144)(112,133,122,143)(113,132,123,142)(114,131,124,141)(115,140,125,150)(116,139,126,149)(117,138,127,148)(118,137,128,147)(119,136,129,146)(120,135,130,145), (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)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(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)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,100,30,110)(2,91,21,101)(3,92,22,102)(4,93,23,103)(5,94,24,104)(6,95,25,105)(7,96,26,106)(8,97,27,107)(9,98,28,108)(10,99,29,109)(11,75,155,85)(12,76,156,86)(13,77,157,87)(14,78,158,88)(15,79,159,89)(16,80,160,90)(17,71,151,81)(18,72,152,82)(19,73,153,83)(20,74,154,84)(31,111,41,121)(32,112,42,122)(33,113,43,123)(34,114,44,124)(35,115,45,125)(36,116,46,126)(37,117,47,127)(38,118,48,128)(39,119,49,129)(40,120,50,130)(51,131,61,141)(52,132,62,142)(53,133,63,143)(54,134,64,144)(55,135,65,145)(56,136,66,146)(57,137,67,147)(58,138,68,148)(59,139,69,149)(60,140,70,150), (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)], [(1,90,30,80),(2,89,21,79),(3,88,22,78),(4,87,23,77),(5,86,24,76),(6,85,25,75),(7,84,26,74),(8,83,27,73),(9,82,28,72),(10,81,29,71),(11,105,155,95),(12,104,156,94),(13,103,157,93),(14,102,158,92),(15,101,159,91),(16,110,160,100),(17,109,151,99),(18,108,152,98),(19,107,153,97),(20,106,154,96),(31,54,41,64),(32,53,42,63),(33,52,43,62),(34,51,44,61),(35,60,45,70),(36,59,46,69),(37,58,47,68),(38,57,48,67),(39,56,49,66),(40,55,50,65),(111,134,121,144),(112,133,122,143),(113,132,123,142),(114,131,124,141),(115,140,125,150),(116,139,126,149),(117,138,127,148),(118,137,128,147),(119,136,129,146),(120,135,130,145)], [(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),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(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),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,100,30,110),(2,91,21,101),(3,92,22,102),(4,93,23,103),(5,94,24,104),(6,95,25,105),(7,96,26,106),(8,97,27,107),(9,98,28,108),(10,99,29,109),(11,75,155,85),(12,76,156,86),(13,77,157,87),(14,78,158,88),(15,79,159,89),(16,80,160,90),(17,71,151,81),(18,72,152,82),(19,73,153,83),(20,74,154,84),(31,111,41,121),(32,112,42,122),(33,113,43,123),(34,114,44,124),(35,115,45,125),(36,116,46,126),(37,117,47,127),(38,118,48,128),(39,119,49,129),(40,120,50,130),(51,131,61,141),(52,132,62,142),(53,133,63,143),(54,134,64,144),(55,135,65,145),(56,136,66,146),(57,137,67,147),(58,138,68,148),(59,139,69,149),(60,140,70,150)], [(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)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| 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 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | C4○D20 | 2+ 1+4 | Q8⋊2D5 | D4⋊6D10 |
| kernel | C10.112+ 1+4 | C4.Dic10 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊D20 | C4⋊C4⋊D5 | C4×C5⋊D4 | C20⋊7D4 | C10×C4⋊C4 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 2 | 4 | 4 | 8 | 6 | 16 | 1 | 4 | 4 |
Matrix representation of C10.112+ 1+4 ►in GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 11 | 9 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 4 |
| 0 | 0 | 20 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 36 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[11,14,0,0,9,30,0,0,0,0,9,0,0,0,0,9],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,20,0,0,4,1],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,36,9] >;
C10.112+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{11}2_+^{1+4} % in TeX
G:=Group("C10.11ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1186);
// by ID
G=gap.SmallGroup(320,1186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,1571,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations