G = C10.202- 1+4 order 320 = 26·5
20th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.202- 1+4, C10.542+ 1+4, C22⋊Q8⋊15D5, C4⋊C4.193D10, (C2×Q8).76D10, D20⋊8C4⋊29C2, D10⋊3Q8⋊20C2, D10⋊D4.1C2, C22⋊C4.18D10, D10.11(C4○D4), (C2×C20).627C23, (C2×C10).182C24, (C22×C4).244D10, D10.12D4⋊26C2, D10.13D4⋊20C2, C2.56(D4⋊6D10), Dic5.Q8⋊20C2, (C2×D20).157C22, C23.D10⋊22C2, C4⋊Dic5.218C22, (Q8×C10).112C22, C22.203(C23×D5), C23.121(C22×D5), C23.23D10⋊24C2, (C22×C10).210C23, (C22×C20).382C22, C5⋊4(C22.33C24), (C2×Dic5).247C23, (C4×Dic5).118C22, C10.D4.80C22, (C22×D5).213C23, C23.D5.122C22, D10⋊C4.148C22, C2.21(Q8.10D10), (D5×C4⋊C4)⋊29C2, (C4×C5⋊D4)⋊58C2, C2.53(D5×C4○D4), C4⋊C4⋊D5⋊18C2, (C5×C22⋊Q8)⋊18C2, C10.165(C2×C4○D4), (C2×C4×D5).261C22, (C2×C4).52(C22×D5), (C5×C4⋊C4).163C22, (C2×C5⋊D4).137C22, (C5×C22⋊C4).37C22, SmallGroup(320,1310)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.202- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=a5b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=a5b2d >
Subgroups: 766 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C23.D10, D10.12D4, D10⋊D4, Dic5.Q8, D5×C4⋊C4, D20⋊8C4, D10.13D4, C4⋊C4⋊D5, C4×C5⋊D4, C23.23D10, D10⋊3Q8, C5×C22⋊Q8, C10.202- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, C23×D5, D4⋊6D10, Q8.10D10, D5×C4○D4, C10.202- 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 70 23 53)(2 69 24 52)(3 68 25 51)(4 67 26 60)(5 66 27 59)(6 65 28 58)(7 64 29 57)(8 63 30 56)(9 62 21 55)(10 61 22 54)(11 125 158 118)(12 124 159 117)(13 123 160 116)(14 122 151 115)(15 121 152 114)(16 130 153 113)(17 129 154 112)(18 128 155 111)(19 127 156 120)(20 126 157 119)(31 75 48 82)(32 74 49 81)(33 73 50 90)(34 72 41 89)(35 71 42 88)(36 80 43 87)(37 79 44 86)(38 78 45 85)(39 77 46 84)(40 76 47 83)(91 147 108 140)(92 146 109 139)(93 145 110 138)(94 144 101 137)(95 143 102 136)(96 142 103 135)(97 141 104 134)(98 150 105 133)(99 149 106 132)(100 148 107 131)
(1 113 6 118)(2 114 7 119)(3 115 8 120)(4 116 9 111)(5 117 10 112)(11 58 16 53)(12 59 17 54)(13 60 18 55)(14 51 19 56)(15 52 20 57)(21 128 26 123)(22 129 27 124)(23 130 28 125)(24 121 29 126)(25 122 30 127)(31 103 36 108)(32 104 37 109)(33 105 38 110)(34 106 39 101)(35 107 40 102)(41 99 46 94)(42 100 47 95)(43 91 48 96)(44 92 49 97)(45 93 50 98)(61 159 66 154)(62 160 67 155)(63 151 68 156)(64 152 69 157)(65 153 70 158)(71 143 76 148)(72 144 77 149)(73 145 78 150)(74 146 79 141)(75 147 80 142)(81 139 86 134)(82 140 87 135)(83 131 88 136)(84 132 89 137)(85 133 90 138)
(1 85 28 73)(2 84 29 72)(3 83 30 71)(4 82 21 80)(5 81 22 79)(6 90 23 78)(7 89 24 77)(8 88 25 76)(9 87 26 75)(10 86 27 74)(11 93 153 105)(12 92 154 104)(13 91 155 103)(14 100 156 102)(15 99 157 101)(16 98 158 110)(17 97 159 109)(18 96 160 108)(19 95 151 107)(20 94 152 106)(31 60 43 62)(32 59 44 61)(33 58 45 70)(34 57 46 69)(35 56 47 68)(36 55 48 67)(37 54 49 66)(38 53 50 65)(39 52 41 64)(40 51 42 63)(111 140 123 142)(112 139 124 141)(113 138 125 150)(114 137 126 149)(115 136 127 148)(116 135 128 147)(117 134 129 146)(118 133 130 145)(119 132 121 144)(120 131 122 143)
(1 125 6 130)(2 124 7 129)(3 123 8 128)(4 122 9 127)(5 121 10 126)(11 58 16 53)(12 57 17 52)(13 56 18 51)(14 55 19 60)(15 54 20 59)(21 120 26 115)(22 119 27 114)(23 118 28 113)(24 117 29 112)(25 116 30 111)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(61 157 66 152)(62 156 67 151)(63 155 68 160)(64 154 69 159)(65 153 70 158)(71 142 76 147)(72 141 77 146)(73 150 78 145)(74 149 79 144)(75 148 80 143)(81 132 86 137)(82 131 87 136)(83 140 88 135)(84 139 89 134)(85 138 90 133)
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), (1,70,23,53)(2,69,24,52)(3,68,25,51)(4,67,26,60)(5,66,27,59)(6,65,28,58)(7,64,29,57)(8,63,30,56)(9,62,21,55)(10,61,22,54)(11,125,158,118)(12,124,159,117)(13,123,160,116)(14,122,151,115)(15,121,152,114)(16,130,153,113)(17,129,154,112)(18,128,155,111)(19,127,156,120)(20,126,157,119)(31,75,48,82)(32,74,49,81)(33,73,50,90)(34,72,41,89)(35,71,42,88)(36,80,43,87)(37,79,44,86)(38,78,45,85)(39,77,46,84)(40,76,47,83)(91,147,108,140)(92,146,109,139)(93,145,110,138)(94,144,101,137)(95,143,102,136)(96,142,103,135)(97,141,104,134)(98,150,105,133)(99,149,106,132)(100,148,107,131), (1,113,6,118)(2,114,7,119)(3,115,8,120)(4,116,9,111)(5,117,10,112)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,51,19,56)(15,52,20,57)(21,128,26,123)(22,129,27,124)(23,130,28,125)(24,121,29,126)(25,122,30,127)(31,103,36,108)(32,104,37,109)(33,105,38,110)(34,106,39,101)(35,107,40,102)(41,99,46,94)(42,100,47,95)(43,91,48,96)(44,92,49,97)(45,93,50,98)(61,159,66,154)(62,160,67,155)(63,151,68,156)(64,152,69,157)(65,153,70,158)(71,143,76,148)(72,144,77,149)(73,145,78,150)(74,146,79,141)(75,147,80,142)(81,139,86,134)(82,140,87,135)(83,131,88,136)(84,132,89,137)(85,133,90,138), (1,85,28,73)(2,84,29,72)(3,83,30,71)(4,82,21,80)(5,81,22,79)(6,90,23,78)(7,89,24,77)(8,88,25,76)(9,87,26,75)(10,86,27,74)(11,93,153,105)(12,92,154,104)(13,91,155,103)(14,100,156,102)(15,99,157,101)(16,98,158,110)(17,97,159,109)(18,96,160,108)(19,95,151,107)(20,94,152,106)(31,60,43,62)(32,59,44,61)(33,58,45,70)(34,57,46,69)(35,56,47,68)(36,55,48,67)(37,54,49,66)(38,53,50,65)(39,52,41,64)(40,51,42,63)(111,140,123,142)(112,139,124,141)(113,138,125,150)(114,137,126,149)(115,136,127,148)(116,135,128,147)(117,134,129,146)(118,133,130,145)(119,132,121,144)(120,131,122,143), (1,125,6,130)(2,124,7,129)(3,123,8,128)(4,122,9,127)(5,121,10,126)(11,58,16,53)(12,57,17,52)(13,56,18,51)(14,55,19,60)(15,54,20,59)(21,120,26,115)(22,119,27,114)(23,118,28,113)(24,117,29,112)(25,116,30,111)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,157,66,152)(62,156,67,151)(63,155,68,160)(64,154,69,159)(65,153,70,158)(71,142,76,147)(72,141,77,146)(73,150,78,145)(74,149,79,144)(75,148,80,143)(81,132,86,137)(82,131,87,136)(83,140,88,135)(84,139,89,134)(85,138,90,133)>;
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), (1,70,23,53)(2,69,24,52)(3,68,25,51)(4,67,26,60)(5,66,27,59)(6,65,28,58)(7,64,29,57)(8,63,30,56)(9,62,21,55)(10,61,22,54)(11,125,158,118)(12,124,159,117)(13,123,160,116)(14,122,151,115)(15,121,152,114)(16,130,153,113)(17,129,154,112)(18,128,155,111)(19,127,156,120)(20,126,157,119)(31,75,48,82)(32,74,49,81)(33,73,50,90)(34,72,41,89)(35,71,42,88)(36,80,43,87)(37,79,44,86)(38,78,45,85)(39,77,46,84)(40,76,47,83)(91,147,108,140)(92,146,109,139)(93,145,110,138)(94,144,101,137)(95,143,102,136)(96,142,103,135)(97,141,104,134)(98,150,105,133)(99,149,106,132)(100,148,107,131), (1,113,6,118)(2,114,7,119)(3,115,8,120)(4,116,9,111)(5,117,10,112)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,51,19,56)(15,52,20,57)(21,128,26,123)(22,129,27,124)(23,130,28,125)(24,121,29,126)(25,122,30,127)(31,103,36,108)(32,104,37,109)(33,105,38,110)(34,106,39,101)(35,107,40,102)(41,99,46,94)(42,100,47,95)(43,91,48,96)(44,92,49,97)(45,93,50,98)(61,159,66,154)(62,160,67,155)(63,151,68,156)(64,152,69,157)(65,153,70,158)(71,143,76,148)(72,144,77,149)(73,145,78,150)(74,146,79,141)(75,147,80,142)(81,139,86,134)(82,140,87,135)(83,131,88,136)(84,132,89,137)(85,133,90,138), (1,85,28,73)(2,84,29,72)(3,83,30,71)(4,82,21,80)(5,81,22,79)(6,90,23,78)(7,89,24,77)(8,88,25,76)(9,87,26,75)(10,86,27,74)(11,93,153,105)(12,92,154,104)(13,91,155,103)(14,100,156,102)(15,99,157,101)(16,98,158,110)(17,97,159,109)(18,96,160,108)(19,95,151,107)(20,94,152,106)(31,60,43,62)(32,59,44,61)(33,58,45,70)(34,57,46,69)(35,56,47,68)(36,55,48,67)(37,54,49,66)(38,53,50,65)(39,52,41,64)(40,51,42,63)(111,140,123,142)(112,139,124,141)(113,138,125,150)(114,137,126,149)(115,136,127,148)(116,135,128,147)(117,134,129,146)(118,133,130,145)(119,132,121,144)(120,131,122,143), (1,125,6,130)(2,124,7,129)(3,123,8,128)(4,122,9,127)(5,121,10,126)(11,58,16,53)(12,57,17,52)(13,56,18,51)(14,55,19,60)(15,54,20,59)(21,120,26,115)(22,119,27,114)(23,118,28,113)(24,117,29,112)(25,116,30,111)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,157,66,152)(62,156,67,151)(63,155,68,160)(64,154,69,159)(65,153,70,158)(71,142,76,147)(72,141,77,146)(73,150,78,145)(74,149,79,144)(75,148,80,143)(81,132,86,137)(82,131,87,136)(83,140,88,135)(84,139,89,134)(85,138,90,133) );
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)], [(1,70,23,53),(2,69,24,52),(3,68,25,51),(4,67,26,60),(5,66,27,59),(6,65,28,58),(7,64,29,57),(8,63,30,56),(9,62,21,55),(10,61,22,54),(11,125,158,118),(12,124,159,117),(13,123,160,116),(14,122,151,115),(15,121,152,114),(16,130,153,113),(17,129,154,112),(18,128,155,111),(19,127,156,120),(20,126,157,119),(31,75,48,82),(32,74,49,81),(33,73,50,90),(34,72,41,89),(35,71,42,88),(36,80,43,87),(37,79,44,86),(38,78,45,85),(39,77,46,84),(40,76,47,83),(91,147,108,140),(92,146,109,139),(93,145,110,138),(94,144,101,137),(95,143,102,136),(96,142,103,135),(97,141,104,134),(98,150,105,133),(99,149,106,132),(100,148,107,131)], [(1,113,6,118),(2,114,7,119),(3,115,8,120),(4,116,9,111),(5,117,10,112),(11,58,16,53),(12,59,17,54),(13,60,18,55),(14,51,19,56),(15,52,20,57),(21,128,26,123),(22,129,27,124),(23,130,28,125),(24,121,29,126),(25,122,30,127),(31,103,36,108),(32,104,37,109),(33,105,38,110),(34,106,39,101),(35,107,40,102),(41,99,46,94),(42,100,47,95),(43,91,48,96),(44,92,49,97),(45,93,50,98),(61,159,66,154),(62,160,67,155),(63,151,68,156),(64,152,69,157),(65,153,70,158),(71,143,76,148),(72,144,77,149),(73,145,78,150),(74,146,79,141),(75,147,80,142),(81,139,86,134),(82,140,87,135),(83,131,88,136),(84,132,89,137),(85,133,90,138)], [(1,85,28,73),(2,84,29,72),(3,83,30,71),(4,82,21,80),(5,81,22,79),(6,90,23,78),(7,89,24,77),(8,88,25,76),(9,87,26,75),(10,86,27,74),(11,93,153,105),(12,92,154,104),(13,91,155,103),(14,100,156,102),(15,99,157,101),(16,98,158,110),(17,97,159,109),(18,96,160,108),(19,95,151,107),(20,94,152,106),(31,60,43,62),(32,59,44,61),(33,58,45,70),(34,57,46,69),(35,56,47,68),(36,55,48,67),(37,54,49,66),(38,53,50,65),(39,52,41,64),(40,51,42,63),(111,140,123,142),(112,139,124,141),(113,138,125,150),(114,137,126,149),(115,136,127,148),(116,135,128,147),(117,134,129,146),(118,133,130,145),(119,132,121,144),(120,131,122,143)], [(1,125,6,130),(2,124,7,129),(3,123,8,128),(4,122,9,127),(5,121,10,126),(11,58,16,53),(12,57,17,52),(13,56,18,51),(14,55,19,60),(15,54,20,59),(21,120,26,115),(22,119,27,114),(23,118,28,113),(24,117,29,112),(25,116,30,111),(31,100,36,95),(32,99,37,94),(33,98,38,93),(34,97,39,92),(35,96,40,91),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(61,157,66,152),(62,156,67,151),(63,155,68,160),(64,154,69,159),(65,153,70,158),(71,142,76,147),(72,141,77,146),(73,150,78,145),(74,149,79,144),(75,148,80,143),(81,132,86,137),(82,131,87,136),(83,140,88,135),(84,139,89,134),(85,138,90,133)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | Q8.10D10 | D5×C4○D4 |
| kernel | C10.202- 1+4 | C23.D10 | D10.12D4 | D10⋊D4 | Dic5.Q8 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | C4⋊C4⋊D5 | C4×C5⋊D4 | C23.23D10 | D10⋊3Q8 | C5×C22⋊Q8 | C22⋊Q8 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.202- 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 | 35 | 0 | 35 | 35 |
| 0 | 0 | 40 | 6 | 6 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 23 | 2 | 29 |
| 0 | 0 | 23 | 3 | 2 | 27 |
| 0 | 0 | 35 | 0 | 18 | 35 |
| 0 | 0 | 38 | 18 | 18 | 35 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 2 | 39 |
| 0 | 0 | 29 | 30 | 4 | 29 |
| 0 | 0 | 33 | 15 | 11 | 0 |
| 0 | 0 | 35 | 15 | 39 | 13 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 18 | 0 | 0 |
| 0 | 0 | 4 | 38 | 0 | 0 |
| 0 | 0 | 35 | 0 | 18 | 35 |
| 0 | 0 | 20 | 23 | 20 | 23 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 10 | 39 |
| 0 | 0 | 34 | 36 | 27 | 29 |
| 0 | 0 | 8 | 26 | 30 | 0 |
| 0 | 0 | 26 | 37 | 6 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,35,40,0,0,35,34,0,6,0,0,0,0,35,6,0,0,0,0,35,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,26,23,35,38,0,0,23,3,0,18,0,0,2,2,18,18,0,0,29,27,35,35],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,28,29,33,35,0,0,0,30,15,15,0,0,2,4,11,39,0,0,39,29,0,13],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,3,4,35,20,0,0,18,38,0,23,0,0,0,0,18,20,0,0,0,0,35,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,3,34,8,26,0,0,11,36,26,37,0,0,10,27,30,6,0,0,39,29,0,13] >;
C10.202- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{20}2_-^{1+4} % in TeX
G:=Group("C10.20ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1310);
// by ID
G=gap.SmallGroup(320,1310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=e^2=a^5,d^2=a^5*b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations