G = C10.392+ 1+4 order 320 = 26·5
39th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.392+ 1+4, C10.722- 1+4, (C4×D5)⋊2D4, C4⋊D4⋊11D5, C4.184(D4×D5), C20⋊2D4⋊19C2, C4⋊C4.180D10, (C2×D4).92D10, D10.15(C2×D4), C20.228(C2×D4), C22⋊C4.8D10, D10⋊D4⋊19C2, D10⋊2Q8⋊21C2, (C2×C20).39C23, Dic5.86(C2×D4), C10.67(C22×D4), Dic5⋊D4⋊13C2, C20.48D4⋊34C2, (C2×C10).152C24, (C22×C4).223D10, C2.41(D4⋊6D10), C23.16(C22×D5), (C2×D20).228C22, (D4×C10).122C22, C4⋊Dic5.207C22, (C2×Dic5).73C23, C22.173(C23×D5), Dic5.14D4⋊19C2, C23.D5.25C22, D10⋊C4.15C22, (C22×C10).187C23, (C22×C20).241C22, C5⋊3(C22.31C24), C10.D4.18C22, (C22×D5).197C23, C2.30(D4.10D10), (C2×Dic10).160C22, (C22×Dic5).109C22, C2.40(C2×D4×D5), (D5×C4⋊C4)⋊21C2, (C2×C4○D20)⋊21C2, (C5×C4⋊D4)⋊14C2, (C2×D4⋊2D5)⋊13C2, (C2×C4×D5).92C22, (C5×C4⋊C4).144C22, (C2×C4).586(C22×D5), (C2×C5⋊D4).28C22, (C5×C22⋊C4).13C22, SmallGroup(320,1280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.392+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >
Subgroups: 1102 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22.31C24, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5.14D4, D10⋊D4, D5×C4⋊C4, D10⋊2Q8, C20.48D4, C20⋊2D4, C20⋊2D4, Dic5⋊D4, C5×C4⋊D4, C2×C4○D20, C2×D4⋊2D5, C10.392+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, 2- 1+4, C22×D5, C22.31C24, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D4.10D10, C10.392+ 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 72)(2 89 21 71)(3 88 22 80)(4 87 23 79)(5 86 24 78)(6 85 25 77)(7 84 26 76)(8 83 27 75)(9 82 28 74)(10 81 29 73)(11 97 157 105)(12 96 158 104)(13 95 159 103)(14 94 160 102)(15 93 151 101)(16 92 152 110)(17 91 153 109)(18 100 154 108)(19 99 155 107)(20 98 156 106)(31 66 49 58)(32 65 50 57)(33 64 41 56)(34 63 42 55)(35 62 43 54)(36 61 44 53)(37 70 45 52)(38 69 46 51)(39 68 47 60)(40 67 48 59)(111 138 129 146)(112 137 130 145)(113 136 121 144)(114 135 122 143)(115 134 123 142)(116 133 124 141)(117 132 125 150)(118 131 126 149)(119 140 127 148)(120 139 128 147)
(1 45 6 50)(2 44 7 49)(3 43 8 48)(4 42 9 47)(5 41 10 46)(11 132 16 137)(12 131 17 136)(13 140 18 135)(14 139 19 134)(15 138 20 133)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)(51 73 56 78)(52 72 57 77)(53 71 58 76)(54 80 59 75)(55 79 60 74)(61 89 66 84)(62 88 67 83)(63 87 68 82)(64 86 69 81)(65 85 70 90)(91 118 96 113)(92 117 97 112)(93 116 98 111)(94 115 99 120)(95 114 100 119)(101 124 106 129)(102 123 107 128)(103 122 108 127)(104 121 109 126)(105 130 110 125)(141 151 146 156)(142 160 147 155)(143 159 148 154)(144 158 149 153)(145 157 150 152)
(1 150 30 132)(2 141 21 133)(3 142 22 134)(4 143 23 135)(5 144 24 136)(6 145 25 137)(7 146 26 138)(8 147 27 139)(9 148 28 140)(10 149 29 131)(11 45 157 37)(12 46 158 38)(13 47 159 39)(14 48 160 40)(15 49 151 31)(16 50 152 32)(17 41 153 33)(18 42 154 34)(19 43 155 35)(20 44 156 36)(51 109 69 91)(52 110 70 92)(53 101 61 93)(54 102 62 94)(55 103 63 95)(56 104 64 96)(57 105 65 97)(58 106 66 98)(59 107 67 99)(60 108 68 100)(71 129 89 111)(72 130 90 112)(73 121 81 113)(74 122 82 114)(75 123 83 115)(76 124 84 116)(77 125 85 117)(78 126 86 118)(79 127 87 119)(80 128 88 120)
(1 37)(2 38)(3 39)(4 40)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 137)(12 138)(13 139)(14 140)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 46)(22 47)(23 48)(24 49)(25 50)(26 41)(27 42)(28 43)(29 44)(30 45)(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)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 121)(99 122)(100 123)(101 118)(102 119)(103 120)(104 111)(105 112)(106 113)(107 114)(108 115)(109 116)(110 117)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 151)(150 152)
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), (1,90,30,72)(2,89,21,71)(3,88,22,80)(4,87,23,79)(5,86,24,78)(6,85,25,77)(7,84,26,76)(8,83,27,75)(9,82,28,74)(10,81,29,73)(11,97,157,105)(12,96,158,104)(13,95,159,103)(14,94,160,102)(15,93,151,101)(16,92,152,110)(17,91,153,109)(18,100,154,108)(19,99,155,107)(20,98,156,106)(31,66,49,58)(32,65,50,57)(33,64,41,56)(34,63,42,55)(35,62,43,54)(36,61,44,53)(37,70,45,52)(38,69,46,51)(39,68,47,60)(40,67,48,59)(111,138,129,146)(112,137,130,145)(113,136,121,144)(114,135,122,143)(115,134,123,142)(116,133,124,141)(117,132,125,150)(118,131,126,149)(119,140,127,148)(120,139,128,147), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,132,16,137)(12,131,17,136)(13,140,18,135)(14,139,19,134)(15,138,20,133)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(51,73,56,78)(52,72,57,77)(53,71,58,76)(54,80,59,75)(55,79,60,74)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,118,96,113)(92,117,97,112)(93,116,98,111)(94,115,99,120)(95,114,100,119)(101,124,106,129)(102,123,107,128)(103,122,108,127)(104,121,109,126)(105,130,110,125)(141,151,146,156)(142,160,147,155)(143,159,148,154)(144,158,149,153)(145,157,150,152), (1,150,30,132)(2,141,21,133)(3,142,22,134)(4,143,23,135)(5,144,24,136)(6,145,25,137)(7,146,26,138)(8,147,27,139)(9,148,28,140)(10,149,29,131)(11,45,157,37)(12,46,158,38)(13,47,159,39)(14,48,160,40)(15,49,151,31)(16,50,152,32)(17,41,153,33)(18,42,154,34)(19,43,155,35)(20,44,156,36)(51,109,69,91)(52,110,70,92)(53,101,61,93)(54,102,62,94)(55,103,63,95)(56,104,64,96)(57,105,65,97)(58,106,66,98)(59,107,67,99)(60,108,68,100)(71,129,89,111)(72,130,90,112)(73,121,81,113)(74,122,82,114)(75,123,83,115)(76,124,84,116)(77,125,85,117)(78,126,86,118)(79,127,87,119)(80,128,88,120), (1,37)(2,38)(3,39)(4,40)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,137)(12,138)(13,139)(14,140)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,121)(99,122)(100,123)(101,118)(102,119)(103,120)(104,111)(105,112)(106,113)(107,114)(108,115)(109,116)(110,117)(141,153)(142,154)(143,155)(144,156)(145,157)(146,158)(147,159)(148,160)(149,151)(150,152)>;
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), (1,90,30,72)(2,89,21,71)(3,88,22,80)(4,87,23,79)(5,86,24,78)(6,85,25,77)(7,84,26,76)(8,83,27,75)(9,82,28,74)(10,81,29,73)(11,97,157,105)(12,96,158,104)(13,95,159,103)(14,94,160,102)(15,93,151,101)(16,92,152,110)(17,91,153,109)(18,100,154,108)(19,99,155,107)(20,98,156,106)(31,66,49,58)(32,65,50,57)(33,64,41,56)(34,63,42,55)(35,62,43,54)(36,61,44,53)(37,70,45,52)(38,69,46,51)(39,68,47,60)(40,67,48,59)(111,138,129,146)(112,137,130,145)(113,136,121,144)(114,135,122,143)(115,134,123,142)(116,133,124,141)(117,132,125,150)(118,131,126,149)(119,140,127,148)(120,139,128,147), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,132,16,137)(12,131,17,136)(13,140,18,135)(14,139,19,134)(15,138,20,133)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(51,73,56,78)(52,72,57,77)(53,71,58,76)(54,80,59,75)(55,79,60,74)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,118,96,113)(92,117,97,112)(93,116,98,111)(94,115,99,120)(95,114,100,119)(101,124,106,129)(102,123,107,128)(103,122,108,127)(104,121,109,126)(105,130,110,125)(141,151,146,156)(142,160,147,155)(143,159,148,154)(144,158,149,153)(145,157,150,152), (1,150,30,132)(2,141,21,133)(3,142,22,134)(4,143,23,135)(5,144,24,136)(6,145,25,137)(7,146,26,138)(8,147,27,139)(9,148,28,140)(10,149,29,131)(11,45,157,37)(12,46,158,38)(13,47,159,39)(14,48,160,40)(15,49,151,31)(16,50,152,32)(17,41,153,33)(18,42,154,34)(19,43,155,35)(20,44,156,36)(51,109,69,91)(52,110,70,92)(53,101,61,93)(54,102,62,94)(55,103,63,95)(56,104,64,96)(57,105,65,97)(58,106,66,98)(59,107,67,99)(60,108,68,100)(71,129,89,111)(72,130,90,112)(73,121,81,113)(74,122,82,114)(75,123,83,115)(76,124,84,116)(77,125,85,117)(78,126,86,118)(79,127,87,119)(80,128,88,120), (1,37)(2,38)(3,39)(4,40)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,137)(12,138)(13,139)(14,140)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,46)(22,47)(23,48)(24,49)(25,50)(26,41)(27,42)(28,43)(29,44)(30,45)(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)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,121)(99,122)(100,123)(101,118)(102,119)(103,120)(104,111)(105,112)(106,113)(107,114)(108,115)(109,116)(110,117)(141,153)(142,154)(143,155)(144,156)(145,157)(146,158)(147,159)(148,160)(149,151)(150,152) );
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)], [(1,90,30,72),(2,89,21,71),(3,88,22,80),(4,87,23,79),(5,86,24,78),(6,85,25,77),(7,84,26,76),(8,83,27,75),(9,82,28,74),(10,81,29,73),(11,97,157,105),(12,96,158,104),(13,95,159,103),(14,94,160,102),(15,93,151,101),(16,92,152,110),(17,91,153,109),(18,100,154,108),(19,99,155,107),(20,98,156,106),(31,66,49,58),(32,65,50,57),(33,64,41,56),(34,63,42,55),(35,62,43,54),(36,61,44,53),(37,70,45,52),(38,69,46,51),(39,68,47,60),(40,67,48,59),(111,138,129,146),(112,137,130,145),(113,136,121,144),(114,135,122,143),(115,134,123,142),(116,133,124,141),(117,132,125,150),(118,131,126,149),(119,140,127,148),(120,139,128,147)], [(1,45,6,50),(2,44,7,49),(3,43,8,48),(4,42,9,47),(5,41,10,46),(11,132,16,137),(12,131,17,136),(13,140,18,135),(14,139,19,134),(15,138,20,133),(21,36,26,31),(22,35,27,40),(23,34,28,39),(24,33,29,38),(25,32,30,37),(51,73,56,78),(52,72,57,77),(53,71,58,76),(54,80,59,75),(55,79,60,74),(61,89,66,84),(62,88,67,83),(63,87,68,82),(64,86,69,81),(65,85,70,90),(91,118,96,113),(92,117,97,112),(93,116,98,111),(94,115,99,120),(95,114,100,119),(101,124,106,129),(102,123,107,128),(103,122,108,127),(104,121,109,126),(105,130,110,125),(141,151,146,156),(142,160,147,155),(143,159,148,154),(144,158,149,153),(145,157,150,152)], [(1,150,30,132),(2,141,21,133),(3,142,22,134),(4,143,23,135),(5,144,24,136),(6,145,25,137),(7,146,26,138),(8,147,27,139),(9,148,28,140),(10,149,29,131),(11,45,157,37),(12,46,158,38),(13,47,159,39),(14,48,160,40),(15,49,151,31),(16,50,152,32),(17,41,153,33),(18,42,154,34),(19,43,155,35),(20,44,156,36),(51,109,69,91),(52,110,70,92),(53,101,61,93),(54,102,62,94),(55,103,63,95),(56,104,64,96),(57,105,65,97),(58,106,66,98),(59,107,67,99),(60,108,68,100),(71,129,89,111),(72,130,90,112),(73,121,81,113),(74,122,82,114),(75,123,83,115),(76,124,84,116),(77,125,85,117),(78,126,86,118),(79,127,87,119),(80,128,88,120)], [(1,37),(2,38),(3,39),(4,40),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,137),(12,138),(13,139),(14,140),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,46),(22,47),(23,48),(24,49),(25,50),(26,41),(27,42),(28,43),(29,44),(30,45),(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),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,121),(99,122),(100,123),(101,118),(102,119),(103,120),(104,111),(105,112),(106,113),(107,114),(108,115),(109,116),(110,117),(141,153),(142,154),(143,155),(144,156),(145,157),(146,158),(147,159),(148,160),(149,151),(150,152)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 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 | 2 | 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 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 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 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4×D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.392+ 1+4 | Dic5.14D4 | D10⋊D4 | D5×C4⋊C4 | D10⋊2Q8 | C20.48D4 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4⋊2D5 | C4×D5 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 6 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.392+ 1+4 ►in GL8(𝔽41)
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 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 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 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 | 1 | 0 | 0 | 39 |
| 0 | 0 | 0 | 0 | 40 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 40 | 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 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 35 | 1 | 0 |
| 0 | 0 | 0 | 0 | 20 | 3 | 20 | 21 |
| 0 | 0 | 0 | 0 | 0 | 3 | 20 | 3 |
| 0 | 0 | 0 | 0 | 21 | 18 | 24 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
G:=sub<GL(8,GF(41))| [7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,39,1,1,40],[1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,40,0,1,0,0,0,0,2,40,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,20,0,21,0,0,0,0,35,3,3,18,0,0,0,0,1,20,20,24,0,0,0,0,0,21,3,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,40,1,0,0,0,0,2,40,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C10.392+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{39}2_+^{1+4} % in TeX
G:=Group("C10.39ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1280);
// by ID
G=gap.SmallGroup(320,1280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,1123,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations