?
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, D10.15(C2×D4), (C2×D4).92D10, 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), (D4×C10).122C22, (C2×D20).228C22, 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
Subgroups: 1102 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×16], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×C10 [×9], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.31C24, C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], D10⋊D4 [×2], D5×C4⋊C4, D10⋊2Q8, C20.48D4, C20⋊2D4, C20⋊2D4 [×2], Dic5⋊D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D4⋊2D5, C10.392+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.31C24, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10, D4.10D10, C10.392+ (1+4)
Generators and relations
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 >
►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 74)(2 89 21 73)(3 88 22 72)(4 87 23 71)(5 86 24 80)(6 85 25 79)(7 84 26 78)(8 83 27 77)(9 82 28 76)(10 81 29 75)(11 99 159 105)(12 98 160 104)(13 97 151 103)(14 96 152 102)(15 95 153 101)(16 94 154 110)(17 93 155 109)(18 92 156 108)(19 91 157 107)(20 100 158 106)(31 68 47 52)(32 67 48 51)(33 66 49 60)(34 65 50 59)(35 64 41 58)(36 63 42 57)(37 62 43 56)(38 61 44 55)(39 70 45 54)(40 69 46 53)(111 132 127 148)(112 131 128 147)(113 140 129 146)(114 139 130 145)(115 138 121 144)(116 137 122 143)(117 136 123 142)(118 135 124 141)(119 134 125 150)(120 133 126 149)
(1 45 6 50)(2 44 7 49)(3 43 8 48)(4 42 9 47)(5 41 10 46)(11 134 16 139)(12 133 17 138)(13 132 18 137)(14 131 19 136)(15 140 20 135)(21 38 26 33)(22 37 27 32)(23 36 28 31)(24 35 29 40)(25 34 30 39)(51 77 56 72)(52 76 57 71)(53 75 58 80)(54 74 59 79)(55 73 60 78)(61 89 66 84)(62 88 67 83)(63 87 68 82)(64 86 69 81)(65 85 70 90)(91 112 96 117)(92 111 97 116)(93 120 98 115)(94 119 99 114)(95 118 100 113)(101 124 106 129)(102 123 107 128)(103 122 108 127)(104 121 109 126)(105 130 110 125)(141 153 146 158)(142 152 147 157)(143 151 148 156)(144 160 149 155)(145 159 150 154)
(1 150 30 134)(2 141 21 135)(3 142 22 136)(4 143 23 137)(5 144 24 138)(6 145 25 139)(7 146 26 140)(8 147 27 131)(9 148 28 132)(10 149 29 133)(11 45 159 39)(12 46 160 40)(13 47 151 31)(14 48 152 32)(15 49 153 33)(16 50 154 34)(17 41 155 35)(18 42 156 36)(19 43 157 37)(20 44 158 38)(51 107 67 91)(52 108 68 92)(53 109 69 93)(54 110 70 94)(55 101 61 95)(56 102 62 96)(57 103 63 97)(58 104 64 98)(59 105 65 99)(60 106 66 100)(71 127 87 111)(72 128 88 112)(73 129 89 113)(74 130 90 114)(75 121 81 115)(76 122 82 116)(77 123 83 117)(78 124 84 118)(79 125 85 119)(80 126 86 120)
(1 39)(2 40)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 139)(12 140)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(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)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 121)(101 120)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)(109 118)(110 119)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 151)(148 152)(149 153)(150 154)
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), (1,90,30,74)(2,89,21,73)(3,88,22,72)(4,87,23,71)(5,86,24,80)(6,85,25,79)(7,84,26,78)(8,83,27,77)(9,82,28,76)(10,81,29,75)(11,99,159,105)(12,98,160,104)(13,97,151,103)(14,96,152,102)(15,95,153,101)(16,94,154,110)(17,93,155,109)(18,92,156,108)(19,91,157,107)(20,100,158,106)(31,68,47,52)(32,67,48,51)(33,66,49,60)(34,65,50,59)(35,64,41,58)(36,63,42,57)(37,62,43,56)(38,61,44,55)(39,70,45,54)(40,69,46,53)(111,132,127,148)(112,131,128,147)(113,140,129,146)(114,139,130,145)(115,138,121,144)(116,137,122,143)(117,136,123,142)(118,135,124,141)(119,134,125,150)(120,133,126,149), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,134,16,139)(12,133,17,138)(13,132,18,137)(14,131,19,136)(15,140,20,135)(21,38,26,33)(22,37,27,32)(23,36,28,31)(24,35,29,40)(25,34,30,39)(51,77,56,72)(52,76,57,71)(53,75,58,80)(54,74,59,79)(55,73,60,78)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,112,96,117)(92,111,97,116)(93,120,98,115)(94,119,99,114)(95,118,100,113)(101,124,106,129)(102,123,107,128)(103,122,108,127)(104,121,109,126)(105,130,110,125)(141,153,146,158)(142,152,147,157)(143,151,148,156)(144,160,149,155)(145,159,150,154), (1,150,30,134)(2,141,21,135)(3,142,22,136)(4,143,23,137)(5,144,24,138)(6,145,25,139)(7,146,26,140)(8,147,27,131)(9,148,28,132)(10,149,29,133)(11,45,159,39)(12,46,160,40)(13,47,151,31)(14,48,152,32)(15,49,153,33)(16,50,154,34)(17,41,155,35)(18,42,156,36)(19,43,157,37)(20,44,158,38)(51,107,67,91)(52,108,68,92)(53,109,69,93)(54,110,70,94)(55,101,61,95)(56,102,62,96)(57,103,63,97)(58,104,64,98)(59,105,65,99)(60,106,66,100)(71,127,87,111)(72,128,88,112)(73,129,89,113)(74,130,90,114)(75,121,81,115)(76,122,82,116)(77,123,83,117)(78,124,84,118)(79,125,85,119)(80,126,86,120), (1,39)(2,40)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,139)(12,140)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(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)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,121)(101,120)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,151)(148,152)(149,153)(150,154)>;
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), (1,90,30,74)(2,89,21,73)(3,88,22,72)(4,87,23,71)(5,86,24,80)(6,85,25,79)(7,84,26,78)(8,83,27,77)(9,82,28,76)(10,81,29,75)(11,99,159,105)(12,98,160,104)(13,97,151,103)(14,96,152,102)(15,95,153,101)(16,94,154,110)(17,93,155,109)(18,92,156,108)(19,91,157,107)(20,100,158,106)(31,68,47,52)(32,67,48,51)(33,66,49,60)(34,65,50,59)(35,64,41,58)(36,63,42,57)(37,62,43,56)(38,61,44,55)(39,70,45,54)(40,69,46,53)(111,132,127,148)(112,131,128,147)(113,140,129,146)(114,139,130,145)(115,138,121,144)(116,137,122,143)(117,136,123,142)(118,135,124,141)(119,134,125,150)(120,133,126,149), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,134,16,139)(12,133,17,138)(13,132,18,137)(14,131,19,136)(15,140,20,135)(21,38,26,33)(22,37,27,32)(23,36,28,31)(24,35,29,40)(25,34,30,39)(51,77,56,72)(52,76,57,71)(53,75,58,80)(54,74,59,79)(55,73,60,78)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,112,96,117)(92,111,97,116)(93,120,98,115)(94,119,99,114)(95,118,100,113)(101,124,106,129)(102,123,107,128)(103,122,108,127)(104,121,109,126)(105,130,110,125)(141,153,146,158)(142,152,147,157)(143,151,148,156)(144,160,149,155)(145,159,150,154), (1,150,30,134)(2,141,21,135)(3,142,22,136)(4,143,23,137)(5,144,24,138)(6,145,25,139)(7,146,26,140)(8,147,27,131)(9,148,28,132)(10,149,29,133)(11,45,159,39)(12,46,160,40)(13,47,151,31)(14,48,152,32)(15,49,153,33)(16,50,154,34)(17,41,155,35)(18,42,156,36)(19,43,157,37)(20,44,158,38)(51,107,67,91)(52,108,68,92)(53,109,69,93)(54,110,70,94)(55,101,61,95)(56,102,62,96)(57,103,63,97)(58,104,64,98)(59,105,65,99)(60,106,66,100)(71,127,87,111)(72,128,88,112)(73,129,89,113)(74,130,90,114)(75,121,81,115)(76,122,82,116)(77,123,83,117)(78,124,84,118)(79,125,85,119)(80,126,86,120), (1,39)(2,40)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,139)(12,140)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(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)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,121)(101,120)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117)(109,118)(110,119)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,151)(148,152)(149,153)(150,154) );
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)], [(1,90,30,74),(2,89,21,73),(3,88,22,72),(4,87,23,71),(5,86,24,80),(6,85,25,79),(7,84,26,78),(8,83,27,77),(9,82,28,76),(10,81,29,75),(11,99,159,105),(12,98,160,104),(13,97,151,103),(14,96,152,102),(15,95,153,101),(16,94,154,110),(17,93,155,109),(18,92,156,108),(19,91,157,107),(20,100,158,106),(31,68,47,52),(32,67,48,51),(33,66,49,60),(34,65,50,59),(35,64,41,58),(36,63,42,57),(37,62,43,56),(38,61,44,55),(39,70,45,54),(40,69,46,53),(111,132,127,148),(112,131,128,147),(113,140,129,146),(114,139,130,145),(115,138,121,144),(116,137,122,143),(117,136,123,142),(118,135,124,141),(119,134,125,150),(120,133,126,149)], [(1,45,6,50),(2,44,7,49),(3,43,8,48),(4,42,9,47),(5,41,10,46),(11,134,16,139),(12,133,17,138),(13,132,18,137),(14,131,19,136),(15,140,20,135),(21,38,26,33),(22,37,27,32),(23,36,28,31),(24,35,29,40),(25,34,30,39),(51,77,56,72),(52,76,57,71),(53,75,58,80),(54,74,59,79),(55,73,60,78),(61,89,66,84),(62,88,67,83),(63,87,68,82),(64,86,69,81),(65,85,70,90),(91,112,96,117),(92,111,97,116),(93,120,98,115),(94,119,99,114),(95,118,100,113),(101,124,106,129),(102,123,107,128),(103,122,108,127),(104,121,109,126),(105,130,110,125),(141,153,146,158),(142,152,147,157),(143,151,148,156),(144,160,149,155),(145,159,150,154)], [(1,150,30,134),(2,141,21,135),(3,142,22,136),(4,143,23,137),(5,144,24,138),(6,145,25,139),(7,146,26,140),(8,147,27,131),(9,148,28,132),(10,149,29,133),(11,45,159,39),(12,46,160,40),(13,47,151,31),(14,48,152,32),(15,49,153,33),(16,50,154,34),(17,41,155,35),(18,42,156,36),(19,43,157,37),(20,44,158,38),(51,107,67,91),(52,108,68,92),(53,109,69,93),(54,110,70,94),(55,101,61,95),(56,102,62,96),(57,103,63,97),(58,104,64,98),(59,105,65,99),(60,106,66,100),(71,127,87,111),(72,128,88,112),(73,129,89,113),(74,130,90,114),(75,121,81,115),(76,122,82,116),(77,123,83,117),(78,124,84,118),(79,125,85,119),(80,126,86,120)], [(1,39),(2,40),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,139),(12,140),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(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),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,121),(101,120),(102,111),(103,112),(104,113),(105,114),(106,115),(107,116),(108,117),(109,118),(110,119),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,151),(148,152),(149,153),(150,154)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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