G = C10.52- 1+4 order 320 = 26·5
5th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.52- 1+4, C4⋊C4.306D10, C4.Dic10⋊9C2, D10⋊2Q8⋊10C2, C4.93(C4○D20), (C2×C10).57C24, Dic5⋊3Q8⋊10C2, C20.195(C4○D4), C20.48D4⋊28C2, (C2×C20).618C23, Dic5.Q8⋊2C2, (C22×C4).182D10, C22.91(C23×D5), C23.21D10⋊4C2, C4⋊Dic5.193C22, (C2×Dic5).18C23, (C4×Dic5).73C22, (C22×D5).15C23, C23.228(C22×D5), C22.24(D4⋊2D5), (C22×C20).219C22, (C22×C10).406C23, C2.8(Q8.10D10), C5⋊1(C22.46C24), C23.23D10.2C2, C23.D5.142C22, D10⋊C4.141C22, (C2×Dic10).148C22, C10.D4.150C22, (C2×C4⋊C4)⋊22D5, (C10×C4⋊C4)⋊19C2, C4⋊C4⋊D5⋊1C2, C4⋊C4⋊7D5⋊9C2, (C4×C5⋊D4).4C2, C2.26(C2×C4○D20), C10.24(C2×C4○D4), (C2×C4×D5).65C22, C2.17(C2×D4⋊2D5), (C5×C4⋊C4).298C22, (C2×C4).145(C22×D5), (C2×C10).173(C4○D4), (C2×C5⋊D4).102C22, SmallGroup(320,1185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.52- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 630 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C22×C20, C22×C20, Dic5⋊3Q8, Dic5.Q8, C4.Dic10, C4⋊C4⋊7D5, D10⋊2Q8, C4⋊C4⋊D5, C20.48D4, C23.21D10, C4×C5⋊D4, C23.23D10, C10×C4⋊C4, C10.52- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, Q8.10D10, C10.52- 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 45 28 33)(2 44 29 32)(3 43 30 31)(4 42 21 40)(5 41 22 39)(6 50 23 38)(7 49 24 37)(8 48 25 36)(9 47 26 35)(10 46 27 34)(11 150 153 138)(12 149 154 137)(13 148 155 136)(14 147 156 135)(15 146 157 134)(16 145 158 133)(17 144 159 132)(18 143 160 131)(19 142 151 140)(20 141 152 139)(51 82 63 80)(52 81 64 79)(53 90 65 78)(54 89 66 77)(55 88 67 76)(56 87 68 75)(57 86 69 74)(58 85 70 73)(59 84 61 72)(60 83 62 71)(91 115 103 127)(92 114 104 126)(93 113 105 125)(94 112 106 124)(95 111 107 123)(96 120 108 122)(97 119 109 121)(98 118 110 130)(99 117 101 129)(100 116 102 128)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 138)(12 139)(13 140)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)(91 128)(92 129)(93 130)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 111)(109 112)(110 113)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
(1 90 28 78)(2 81 29 79)(3 82 30 80)(4 83 21 71)(5 84 22 72)(6 85 23 73)(7 86 24 74)(8 87 25 75)(9 88 26 76)(10 89 27 77)(11 105 153 93)(12 106 154 94)(13 107 155 95)(14 108 156 96)(15 109 157 97)(16 110 158 98)(17 101 159 99)(18 102 160 100)(19 103 151 91)(20 104 152 92)(31 56 43 68)(32 57 44 69)(33 58 45 70)(34 59 46 61)(35 60 47 62)(36 51 48 63)(37 52 49 64)(38 53 50 65)(39 54 41 66)(40 55 42 67)(111 143 123 131)(112 144 124 132)(113 145 125 133)(114 146 126 134)(115 147 127 135)(116 148 128 136)(117 149 129 137)(118 150 130 138)(119 141 121 139)(120 142 122 140)
(1 98 23 105)(2 99 24 106)(3 100 25 107)(4 91 26 108)(5 92 27 109)(6 93 28 110)(7 94 29 101)(8 95 30 102)(9 96 21 103)(10 97 22 104)(11 85 158 78)(12 86 159 79)(13 87 160 80)(14 88 151 71)(15 89 152 72)(16 90 153 73)(17 81 154 74)(18 82 155 75)(19 83 156 76)(20 84 157 77)(31 128 48 111)(32 129 49 112)(33 130 50 113)(34 121 41 114)(35 122 42 115)(36 123 43 116)(37 124 44 117)(38 125 45 118)(39 126 46 119)(40 127 47 120)(51 148 68 131)(52 149 69 132)(53 150 70 133)(54 141 61 134)(55 142 62 135)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 146 66 139)(60 147 67 140)
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), (1,45,28,33)(2,44,29,32)(3,43,30,31)(4,42,21,40)(5,41,22,39)(6,50,23,38)(7,49,24,37)(8,48,25,36)(9,47,26,35)(10,46,27,34)(11,150,153,138)(12,149,154,137)(13,148,155,136)(14,147,156,135)(15,146,157,134)(16,145,158,133)(17,144,159,132)(18,143,160,131)(19,142,151,140)(20,141,152,139)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,115,103,127)(92,114,104,126)(93,113,105,125)(94,112,106,124)(95,111,107,123)(96,120,108,122)(97,119,109,121)(98,118,110,130)(99,117,101,129)(100,116,102,128), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,41)(30,42)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,90,28,78)(2,81,29,79)(3,82,30,80)(4,83,21,71)(5,84,22,72)(6,85,23,73)(7,86,24,74)(8,87,25,75)(9,88,26,76)(10,89,27,77)(11,105,153,93)(12,106,154,94)(13,107,155,95)(14,108,156,96)(15,109,157,97)(16,110,158,98)(17,101,159,99)(18,102,160,100)(19,103,151,91)(20,104,152,92)(31,56,43,68)(32,57,44,69)(33,58,45,70)(34,59,46,61)(35,60,47,62)(36,51,48,63)(37,52,49,64)(38,53,50,65)(39,54,41,66)(40,55,42,67)(111,143,123,131)(112,144,124,132)(113,145,125,133)(114,146,126,134)(115,147,127,135)(116,148,128,136)(117,149,129,137)(118,150,130,138)(119,141,121,139)(120,142,122,140), (1,98,23,105)(2,99,24,106)(3,100,25,107)(4,91,26,108)(5,92,27,109)(6,93,28,110)(7,94,29,101)(8,95,30,102)(9,96,21,103)(10,97,22,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140)>;
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), (1,45,28,33)(2,44,29,32)(3,43,30,31)(4,42,21,40)(5,41,22,39)(6,50,23,38)(7,49,24,37)(8,48,25,36)(9,47,26,35)(10,46,27,34)(11,150,153,138)(12,149,154,137)(13,148,155,136)(14,147,156,135)(15,146,157,134)(16,145,158,133)(17,144,159,132)(18,143,160,131)(19,142,151,140)(20,141,152,139)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,115,103,127)(92,114,104,126)(93,113,105,125)(94,112,106,124)(95,111,107,123)(96,120,108,122)(97,119,109,121)(98,118,110,130)(99,117,101,129)(100,116,102,128), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,41)(30,42)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,90,28,78)(2,81,29,79)(3,82,30,80)(4,83,21,71)(5,84,22,72)(6,85,23,73)(7,86,24,74)(8,87,25,75)(9,88,26,76)(10,89,27,77)(11,105,153,93)(12,106,154,94)(13,107,155,95)(14,108,156,96)(15,109,157,97)(16,110,158,98)(17,101,159,99)(18,102,160,100)(19,103,151,91)(20,104,152,92)(31,56,43,68)(32,57,44,69)(33,58,45,70)(34,59,46,61)(35,60,47,62)(36,51,48,63)(37,52,49,64)(38,53,50,65)(39,54,41,66)(40,55,42,67)(111,143,123,131)(112,144,124,132)(113,145,125,133)(114,146,126,134)(115,147,127,135)(116,148,128,136)(117,149,129,137)(118,150,130,138)(119,141,121,139)(120,142,122,140), (1,98,23,105)(2,99,24,106)(3,100,25,107)(4,91,26,108)(5,92,27,109)(6,93,28,110)(7,94,29,101)(8,95,30,102)(9,96,21,103)(10,97,22,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140) );
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)], [(1,45,28,33),(2,44,29,32),(3,43,30,31),(4,42,21,40),(5,41,22,39),(6,50,23,38),(7,49,24,37),(8,48,25,36),(9,47,26,35),(10,46,27,34),(11,150,153,138),(12,149,154,137),(13,148,155,136),(14,147,156,135),(15,146,157,134),(16,145,158,133),(17,144,159,132),(18,143,160,131),(19,142,151,140),(20,141,152,139),(51,82,63,80),(52,81,64,79),(53,90,65,78),(54,89,66,77),(55,88,67,76),(56,87,68,75),(57,86,69,74),(58,85,70,73),(59,84,61,72),(60,83,62,71),(91,115,103,127),(92,114,104,126),(93,113,105,125),(94,112,106,124),(95,111,107,123),(96,120,108,122),(97,119,109,121),(98,118,110,130),(99,117,101,129),(100,116,102,128)], [(1,38),(2,39),(3,40),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,138),(12,139),(13,140),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,41),(30,42),(51,88),(52,89),(53,90),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73),(91,128),(92,129),(93,130),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,111),(109,112),(110,113),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,151),(149,152),(150,153)], [(1,90,28,78),(2,81,29,79),(3,82,30,80),(4,83,21,71),(5,84,22,72),(6,85,23,73),(7,86,24,74),(8,87,25,75),(9,88,26,76),(10,89,27,77),(11,105,153,93),(12,106,154,94),(13,107,155,95),(14,108,156,96),(15,109,157,97),(16,110,158,98),(17,101,159,99),(18,102,160,100),(19,103,151,91),(20,104,152,92),(31,56,43,68),(32,57,44,69),(33,58,45,70),(34,59,46,61),(35,60,47,62),(36,51,48,63),(37,52,49,64),(38,53,50,65),(39,54,41,66),(40,55,42,67),(111,143,123,131),(112,144,124,132),(113,145,125,133),(114,146,126,134),(115,147,127,135),(116,148,128,136),(117,149,129,137),(118,150,130,138),(119,141,121,139),(120,142,122,140)], [(1,98,23,105),(2,99,24,106),(3,100,25,107),(4,91,26,108),(5,92,27,109),(6,93,28,110),(7,94,29,101),(8,95,30,102),(9,96,21,103),(10,97,22,104),(11,85,158,78),(12,86,159,79),(13,87,160,80),(14,88,151,71),(15,89,152,72),(16,90,153,73),(17,81,154,74),(18,82,155,75),(19,83,156,76),(20,84,157,77),(31,128,48,111),(32,129,49,112),(33,130,50,113),(34,121,41,114),(35,122,42,115),(36,123,43,116),(37,124,44,117),(38,125,45,118),(39,126,46,119),(40,127,47,120),(51,148,68,131),(52,149,69,132),(53,150,70,133),(54,141,61,134),(55,142,62,135),(56,143,63,136),(57,144,64,137),(58,145,65,138),(59,146,66,139),(60,147,67,140)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | C4○D20 | 2- 1+4 | D4⋊2D5 | Q8.10D10 |
| kernel | C10.52- 1+4 | Dic5⋊3Q8 | Dic5.Q8 | C4.Dic10 | C4⋊C4⋊7D5 | D10⋊2Q8 | C4⋊C4⋊D5 | C20.48D4 | C23.21D10 | C4×C5⋊D4 | C23.23D10 | C10×C4⋊C4 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 6 | 16 | 1 | 4 | 4 |
Matrix representation of C10.52- 1+4 ►in GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 32 | 1 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 9 | 0 | 0 |
| 32 | 30 | 0 | 0 |
| 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 40 | 32 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[17,3,0,0,40,24,0,0,0,0,40,32,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[11,32,0,0,9,30,0,0,0,0,1,0,0,0,18,40],[40,0,0,0,0,40,0,0,0,0,9,40,0,0,0,32] >;
C10.52- 1+4 in GAP, Magma, Sage, TeX
C_{10}._52_-^{1+4} % in TeX
G:=Group("C10.5ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1185);
// by ID
G=gap.SmallGroup(320,1185);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations