G = C10.742- 1+4 order 320 = 26·5
29th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.742- 1+4, C10.492+ 1+4, C4⋊D4⋊24D5, C4⋊C4.93D10, C20⋊2D4⋊29C2, (C2×D4).96D10, D10⋊Q8⋊16C2, (C2×C20).47C23, C22⋊C4.12D10, Dic5⋊D4⋊19C2, C20.48D4⋊45C2, C20.17D4⋊20C2, (C2×C10).165C24, (C22×C4).231D10, C4⋊Dic5.46C22, D10.12D4⋊23C2, C2.51(D4⋊6D10), Dic5.Q8⋊14C2, Dic5.5D4⋊23C2, (D4×C10).129C22, (C2×Dic5).82C23, (C22×D5).72C23, C23.115(C22×D5), C22.186(C23×D5), Dic5.14D4⋊21C2, C23.D5.29C22, D10⋊C4.18C22, C23.18D10⋊23C2, C23.23D10⋊13C2, (C22×C10).193C23, (C22×C20).313C22, C5⋊1(C22.56C24), (C4×Dic5).108C22, (C2×Dic10).36C22, C10.D4.79C22, C2.32(D4.10D10), (C22×Dic5).116C22, (C5×C4⋊D4)⋊27C2, (C2×C4×D5).99C22, (C2×C4).43(C22×D5), (C5×C4⋊C4).151C22, (C2×C5⋊D4).36C22, (C5×C22⋊C4).20C22, SmallGroup(320,1293)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.742- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=a5b2d >
Subgroups: 790 in 220 conjugacy classes, 91 normal (all 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×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.56C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Dic5.14D4, D10.12D4, Dic5.5D4, Dic5.Q8, D10⋊Q8, C20.48D4, C23.23D10, C23.18D10, C20.17D4, C20⋊2D4, Dic5⋊D4, C5×C4⋊D4, C10.742- 1+4
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.56C24, C23×D5, D4⋊6D10, D4.10D10, C10.742- 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 50 25 33)(2 49 26 32)(3 48 27 31)(4 47 28 40)(5 46 29 39)(6 45 30 38)(7 44 21 37)(8 43 22 36)(9 42 23 35)(10 41 24 34)(11 138 158 145)(12 137 159 144)(13 136 160 143)(14 135 151 142)(15 134 152 141)(16 133 153 150)(17 132 154 149)(18 131 155 148)(19 140 156 147)(20 139 157 146)(51 87 68 80)(52 86 69 79)(53 85 70 78)(54 84 61 77)(55 83 62 76)(56 82 63 75)(57 81 64 74)(58 90 65 73)(59 89 66 72)(60 88 67 71)(91 115 108 122)(92 114 109 121)(93 113 110 130)(94 112 101 129)(95 111 102 128)(96 120 103 127)(97 119 104 126)(98 118 105 125)(99 117 106 124)(100 116 107 123)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(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 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 81)(60 82)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 121)(100 122)(101 119)(102 120)(103 111)(104 112)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 90 25 73)(2 89 26 72)(3 88 27 71)(4 87 28 80)(5 86 29 79)(6 85 30 78)(7 84 21 77)(8 83 22 76)(9 82 23 75)(10 81 24 74)(11 93 158 110)(12 92 159 109)(13 91 160 108)(14 100 151 107)(15 99 152 106)(16 98 153 105)(17 97 154 104)(18 96 155 103)(19 95 156 102)(20 94 157 101)(31 60 48 67)(32 59 49 66)(33 58 50 65)(34 57 41 64)(35 56 42 63)(36 55 43 62)(37 54 44 61)(38 53 45 70)(39 52 46 69)(40 51 47 68)(111 147 128 140)(112 146 129 139)(113 145 130 138)(114 144 121 137)(115 143 122 136)(116 142 123 135)(117 141 124 134)(118 150 125 133)(119 149 126 132)(120 148 127 131)
(1 11 25 158)(2 12 26 159)(3 13 27 160)(4 14 28 151)(5 15 29 152)(6 16 30 153)(7 17 21 154)(8 18 22 155)(9 19 23 156)(10 20 24 157)(31 148 48 131)(32 149 49 132)(33 150 50 133)(34 141 41 134)(35 142 42 135)(36 143 43 136)(37 144 44 137)(38 145 45 138)(39 146 46 139)(40 147 47 140)(51 116 68 123)(52 117 69 124)(53 118 70 125)(54 119 61 126)(55 120 62 127)(56 111 63 128)(57 112 64 129)(58 113 65 130)(59 114 66 121)(60 115 67 122)(71 96 88 103)(72 97 89 104)(73 98 90 105)(74 99 81 106)(75 100 82 107)(76 91 83 108)(77 92 84 109)(78 93 85 110)(79 94 86 101)(80 95 87 102)
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), (1,50,25,33)(2,49,26,32)(3,48,27,31)(4,47,28,40)(5,46,29,39)(6,45,30,38)(7,44,21,37)(8,43,22,36)(9,42,23,35)(10,41,24,34)(11,138,158,145)(12,137,159,144)(13,136,160,143)(14,135,151,142)(15,134,152,141)(16,133,153,150)(17,132,154,149)(18,131,155,148)(19,140,156,147)(20,139,157,146)(51,87,68,80)(52,86,69,79)(53,85,70,78)(54,84,61,77)(55,83,62,76)(56,82,63,75)(57,81,64,74)(58,90,65,73)(59,89,66,72)(60,88,67,71)(91,115,108,122)(92,114,109,121)(93,113,110,130)(94,112,101,129)(95,111,102,128)(96,120,103,127)(97,119,104,126)(98,118,105,125)(99,117,106,124)(100,116,107,123), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(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)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,81)(60,82)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,121)(100,122)(101,119)(102,120)(103,111)(104,112)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,90,25,73)(2,89,26,72)(3,88,27,71)(4,87,28,80)(5,86,29,79)(6,85,30,78)(7,84,21,77)(8,83,22,76)(9,82,23,75)(10,81,24,74)(11,93,158,110)(12,92,159,109)(13,91,160,108)(14,100,151,107)(15,99,152,106)(16,98,153,105)(17,97,154,104)(18,96,155,103)(19,95,156,102)(20,94,157,101)(31,60,48,67)(32,59,49,66)(33,58,50,65)(34,57,41,64)(35,56,42,63)(36,55,43,62)(37,54,44,61)(38,53,45,70)(39,52,46,69)(40,51,47,68)(111,147,128,140)(112,146,129,139)(113,145,130,138)(114,144,121,137)(115,143,122,136)(116,142,123,135)(117,141,124,134)(118,150,125,133)(119,149,126,132)(120,148,127,131), (1,11,25,158)(2,12,26,159)(3,13,27,160)(4,14,28,151)(5,15,29,152)(6,16,30,153)(7,17,21,154)(8,18,22,155)(9,19,23,156)(10,20,24,157)(31,148,48,131)(32,149,49,132)(33,150,50,133)(34,141,41,134)(35,142,42,135)(36,143,43,136)(37,144,44,137)(38,145,45,138)(39,146,46,139)(40,147,47,140)(51,116,68,123)(52,117,69,124)(53,118,70,125)(54,119,61,126)(55,120,62,127)(56,111,63,128)(57,112,64,129)(58,113,65,130)(59,114,66,121)(60,115,67,122)(71,96,88,103)(72,97,89,104)(73,98,90,105)(74,99,81,106)(75,100,82,107)(76,91,83,108)(77,92,84,109)(78,93,85,110)(79,94,86,101)(80,95,87,102)>;
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), (1,50,25,33)(2,49,26,32)(3,48,27,31)(4,47,28,40)(5,46,29,39)(6,45,30,38)(7,44,21,37)(8,43,22,36)(9,42,23,35)(10,41,24,34)(11,138,158,145)(12,137,159,144)(13,136,160,143)(14,135,151,142)(15,134,152,141)(16,133,153,150)(17,132,154,149)(18,131,155,148)(19,140,156,147)(20,139,157,146)(51,87,68,80)(52,86,69,79)(53,85,70,78)(54,84,61,77)(55,83,62,76)(56,82,63,75)(57,81,64,74)(58,90,65,73)(59,89,66,72)(60,88,67,71)(91,115,108,122)(92,114,109,121)(93,113,110,130)(94,112,101,129)(95,111,102,128)(96,120,103,127)(97,119,104,126)(98,118,105,125)(99,117,106,124)(100,116,107,123), (1,38)(2,39)(3,40)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(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)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,81)(60,82)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,121)(100,122)(101,119)(102,120)(103,111)(104,112)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,90,25,73)(2,89,26,72)(3,88,27,71)(4,87,28,80)(5,86,29,79)(6,85,30,78)(7,84,21,77)(8,83,22,76)(9,82,23,75)(10,81,24,74)(11,93,158,110)(12,92,159,109)(13,91,160,108)(14,100,151,107)(15,99,152,106)(16,98,153,105)(17,97,154,104)(18,96,155,103)(19,95,156,102)(20,94,157,101)(31,60,48,67)(32,59,49,66)(33,58,50,65)(34,57,41,64)(35,56,42,63)(36,55,43,62)(37,54,44,61)(38,53,45,70)(39,52,46,69)(40,51,47,68)(111,147,128,140)(112,146,129,139)(113,145,130,138)(114,144,121,137)(115,143,122,136)(116,142,123,135)(117,141,124,134)(118,150,125,133)(119,149,126,132)(120,148,127,131), (1,11,25,158)(2,12,26,159)(3,13,27,160)(4,14,28,151)(5,15,29,152)(6,16,30,153)(7,17,21,154)(8,18,22,155)(9,19,23,156)(10,20,24,157)(31,148,48,131)(32,149,49,132)(33,150,50,133)(34,141,41,134)(35,142,42,135)(36,143,43,136)(37,144,44,137)(38,145,45,138)(39,146,46,139)(40,147,47,140)(51,116,68,123)(52,117,69,124)(53,118,70,125)(54,119,61,126)(55,120,62,127)(56,111,63,128)(57,112,64,129)(58,113,65,130)(59,114,66,121)(60,115,67,122)(71,96,88,103)(72,97,89,104)(73,98,90,105)(74,99,81,106)(75,100,82,107)(76,91,83,108)(77,92,84,109)(78,93,85,110)(79,94,86,101)(80,95,87,102) );
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)], [(1,50,25,33),(2,49,26,32),(3,48,27,31),(4,47,28,40),(5,46,29,39),(6,45,30,38),(7,44,21,37),(8,43,22,36),(9,42,23,35),(10,41,24,34),(11,138,158,145),(12,137,159,144),(13,136,160,143),(14,135,151,142),(15,134,152,141),(16,133,153,150),(17,132,154,149),(18,131,155,148),(19,140,156,147),(20,139,157,146),(51,87,68,80),(52,86,69,79),(53,85,70,78),(54,84,61,77),(55,83,62,76),(56,82,63,75),(57,81,64,74),(58,90,65,73),(59,89,66,72),(60,88,67,71),(91,115,108,122),(92,114,109,121),(93,113,110,130),(94,112,101,129),(95,111,102,128),(96,120,103,127),(97,119,104,126),(98,118,105,125),(99,117,106,124),(100,116,107,123)], [(1,38),(2,39),(3,40),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(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),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,81),(60,82),(61,79),(62,80),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,121),(100,122),(101,119),(102,120),(103,111),(104,112),(105,113),(106,114),(107,115),(108,116),(109,117),(110,118),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,90,25,73),(2,89,26,72),(3,88,27,71),(4,87,28,80),(5,86,29,79),(6,85,30,78),(7,84,21,77),(8,83,22,76),(9,82,23,75),(10,81,24,74),(11,93,158,110),(12,92,159,109),(13,91,160,108),(14,100,151,107),(15,99,152,106),(16,98,153,105),(17,97,154,104),(18,96,155,103),(19,95,156,102),(20,94,157,101),(31,60,48,67),(32,59,49,66),(33,58,50,65),(34,57,41,64),(35,56,42,63),(36,55,43,62),(37,54,44,61),(38,53,45,70),(39,52,46,69),(40,51,47,68),(111,147,128,140),(112,146,129,139),(113,145,130,138),(114,144,121,137),(115,143,122,136),(116,142,123,135),(117,141,124,134),(118,150,125,133),(119,149,126,132),(120,148,127,131)], [(1,11,25,158),(2,12,26,159),(3,13,27,160),(4,14,28,151),(5,15,29,152),(6,16,30,153),(7,17,21,154),(8,18,22,155),(9,19,23,156),(10,20,24,157),(31,148,48,131),(32,149,49,132),(33,150,50,133),(34,141,41,134),(35,142,42,135),(36,143,43,136),(37,144,44,137),(38,145,45,138),(39,146,46,139),(40,147,47,140),(51,116,68,123),(52,117,69,124),(53,118,70,125),(54,119,61,126),(55,120,62,127),(56,111,63,128),(57,112,64,129),(58,113,65,130),(59,114,66,121),(60,115,67,122),(71,96,88,103),(72,97,89,104),(73,98,90,105),(74,99,81,106),(75,100,82,107),(76,91,83,108),(77,92,84,109),(78,93,85,110),(79,94,86,101),(80,95,87,102)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 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 | 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 | 20 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | D4.10D10 |
| kernel | C10.742- 1+4 | Dic5.14D4 | D10.12D4 | Dic5.5D4 | Dic5.Q8 | D10⋊Q8 | C20.48D4 | C23.23D10 | C23.18D10 | C20.17D4 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 6 | 2 | 1 | 8 | 4 |
Matrix representation of C10.742- 1+4 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 9 | 40 | 35 |
| 0 | 0 | 0 | 0 | 2 | 19 | 26 | 1 |
| 0 | 0 | 0 | 0 | 1 | 6 | 19 | 32 |
| 0 | 0 | 0 | 0 | 15 | 40 | 39 | 22 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 24 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,7],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,22,2,1,15,0,0,0,0,9,19,6,40,0,0,0,0,40,26,19,39,0,0,0,0,35,1,32,22],[0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24] >;
C10.742- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{74}2_-^{1+4} % in TeX
G:=Group("C10.74ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1293);
// by ID
G=gap.SmallGroup(320,1293);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,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