?
G = C10.702- (1+4) order 320 = 26·5
25th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.702- (1+4), C10.352+ (1+4), C4⋊C4.91D10, C4⋊D4.8D5, (D4×Dic5)⋊18C2, C22⋊C4.6D10, (C2×D4).154D10, C20.48D4⋊43C2, (C2×C10).146C24, (C2×C20).625C23, (C22×C4).221D10, C4⋊Dic5.45C22, C2.37(D4⋊6D10), C23.12(C22×D5), Dic5.Q8⋊12C2, (D4×C10).120C22, C22.6(D4⋊2D5), C23.D10⋊16C2, (C22×C10).17C23, (C2×Dic5).67C23, C22.167(C23×D5), Dic5.14D4⋊17C2, C23.D5.23C22, C23.18D10⋊21C2, (C22×C20).311C22, C5⋊3(C22.33C24), (C4×Dic5).101C22, (C2×Dic10).35C22, C2.28(D4.10D10), C10.D4.159C22, (C22×Dic5).107C22, C10.82(C2×C4○D4), (C5×C4⋊D4).8C2, C2.34(C2×D4⋊2D5), (C2×C10).22(C4○D4), (C2×C10.D4)⋊29C2, (C5×C4⋊C4).142C22, (C2×C4).174(C22×D5), (C5×C22⋊C4).11C22, SmallGroup(320,1274)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 670 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×14], D4 [×5], Q8, C23, C23 [×2], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], Dic10, C2×Dic5 [×8], C2×Dic5 [×5], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×C10, C22×C10 [×2], C22.33C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], C23.D10 [×2], Dic5.Q8 [×2], C2×C10.D4, C20.48D4, D4×Dic5 [×2], C23.18D10 [×4], C5×C4⋊D4, C10.702- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.33C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊6D10, D4.10D10, C10.702- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >
►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 86 30 73)(2 87 21 74)(3 88 22 75)(4 89 23 76)(5 90 24 77)(6 81 25 78)(7 82 26 79)(8 83 27 80)(9 84 28 71)(10 85 29 72)(11 106 153 93)(12 107 154 94)(13 108 155 95)(14 109 156 96)(15 110 157 97)(16 101 158 98)(17 102 159 99)(18 103 160 100)(19 104 151 91)(20 105 152 92)(31 51 43 63)(32 52 44 64)(33 53 45 65)(34 54 46 66)(35 55 47 67)(36 56 48 68)(37 57 49 69)(38 58 50 70)(39 59 41 61)(40 60 42 62)(111 144 124 131)(112 145 125 132)(113 146 126 133)(114 147 127 134)(115 148 128 135)(116 149 129 136)(117 150 130 137)(118 141 121 138)(119 142 122 139)(120 143 123 140)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 61)(8 62)(9 63)(10 64)(11 126)(12 127)(13 128)(14 129)(15 130)(16 121)(17 122)(18 123)(19 124)(20 125)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 51)(29 52)(30 53)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 81)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 132 25 150)(2 131 26 149)(3 140 27 148)(4 139 28 147)(5 138 29 146)(6 137 30 145)(7 136 21 144)(8 135 22 143)(9 134 23 142)(10 133 24 141)(11 49 158 32)(12 48 159 31)(13 47 160 40)(14 46 151 39)(15 45 152 38)(16 44 153 37)(17 43 154 36)(18 42 155 35)(19 41 156 34)(20 50 157 33)(51 107 68 99)(52 106 69 98)(53 105 70 97)(54 104 61 96)(55 103 62 95)(56 102 63 94)(57 101 64 93)(58 110 65 92)(59 109 66 91)(60 108 67 100)(71 127 89 119)(72 126 90 118)(73 125 81 117)(74 124 82 116)(75 123 83 115)(76 122 84 114)(77 121 85 113)(78 130 86 112)(79 129 87 111)(80 128 88 120)
(1 130 30 117)(2 129 21 116)(3 128 22 115)(4 127 23 114)(5 126 24 113)(6 125 25 112)(7 124 26 111)(8 123 27 120)(9 122 28 119)(10 121 29 118)(11 57 153 69)(12 56 154 68)(13 55 155 67)(14 54 156 66)(15 53 157 65)(16 52 158 64)(17 51 159 63)(18 60 160 62)(19 59 151 61)(20 58 152 70)(31 107 43 94)(32 106 44 93)(33 105 45 92)(34 104 46 91)(35 103 47 100)(36 102 48 99)(37 101 49 98)(38 110 50 97)(39 109 41 96)(40 108 42 95)(71 147 84 134)(72 146 85 133)(73 145 86 132)(74 144 87 131)(75 143 88 140)(76 142 89 139)(77 141 90 138)(78 150 81 137)(79 149 82 136)(80 148 83 135)
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), (1,86,30,73)(2,87,21,74)(3,88,22,75)(4,89,23,76)(5,90,24,77)(6,81,25,78)(7,82,26,79)(8,83,27,80)(9,84,28,71)(10,85,29,72)(11,106,153,93)(12,107,154,94)(13,108,155,95)(14,109,156,96)(15,110,157,97)(16,101,158,98)(17,102,159,99)(18,103,160,100)(19,104,151,91)(20,105,152,92)(31,51,43,63)(32,52,44,64)(33,53,45,65)(34,54,46,66)(35,55,47,67)(36,56,48,68)(37,57,49,69)(38,58,50,70)(39,59,41,61)(40,60,42,62)(111,144,124,131)(112,145,125,132)(113,146,126,133)(114,147,127,134)(115,148,128,135)(116,149,129,136)(117,150,130,137)(118,141,121,138)(119,142,122,139)(120,143,123,140), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,61)(8,62)(9,63)(10,64)(11,126)(12,127)(13,128)(14,129)(15,130)(16,121)(17,122)(18,123)(19,124)(20,125)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,81)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,132,25,150)(2,131,26,149)(3,140,27,148)(4,139,28,147)(5,138,29,146)(6,137,30,145)(7,136,21,144)(8,135,22,143)(9,134,23,142)(10,133,24,141)(11,49,158,32)(12,48,159,31)(13,47,160,40)(14,46,151,39)(15,45,152,38)(16,44,153,37)(17,43,154,36)(18,42,155,35)(19,41,156,34)(20,50,157,33)(51,107,68,99)(52,106,69,98)(53,105,70,97)(54,104,61,96)(55,103,62,95)(56,102,63,94)(57,101,64,93)(58,110,65,92)(59,109,66,91)(60,108,67,100)(71,127,89,119)(72,126,90,118)(73,125,81,117)(74,124,82,116)(75,123,83,115)(76,122,84,114)(77,121,85,113)(78,130,86,112)(79,129,87,111)(80,128,88,120), (1,130,30,117)(2,129,21,116)(3,128,22,115)(4,127,23,114)(5,126,24,113)(6,125,25,112)(7,124,26,111)(8,123,27,120)(9,122,28,119)(10,121,29,118)(11,57,153,69)(12,56,154,68)(13,55,155,67)(14,54,156,66)(15,53,157,65)(16,52,158,64)(17,51,159,63)(18,60,160,62)(19,59,151,61)(20,58,152,70)(31,107,43,94)(32,106,44,93)(33,105,45,92)(34,104,46,91)(35,103,47,100)(36,102,48,99)(37,101,49,98)(38,110,50,97)(39,109,41,96)(40,108,42,95)(71,147,84,134)(72,146,85,133)(73,145,86,132)(74,144,87,131)(75,143,88,140)(76,142,89,139)(77,141,90,138)(78,150,81,137)(79,149,82,136)(80,148,83,135)>;
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), (1,86,30,73)(2,87,21,74)(3,88,22,75)(4,89,23,76)(5,90,24,77)(6,81,25,78)(7,82,26,79)(8,83,27,80)(9,84,28,71)(10,85,29,72)(11,106,153,93)(12,107,154,94)(13,108,155,95)(14,109,156,96)(15,110,157,97)(16,101,158,98)(17,102,159,99)(18,103,160,100)(19,104,151,91)(20,105,152,92)(31,51,43,63)(32,52,44,64)(33,53,45,65)(34,54,46,66)(35,55,47,67)(36,56,48,68)(37,57,49,69)(38,58,50,70)(39,59,41,61)(40,60,42,62)(111,144,124,131)(112,145,125,132)(113,146,126,133)(114,147,127,134)(115,148,128,135)(116,149,129,136)(117,150,130,137)(118,141,121,138)(119,142,122,139)(120,143,123,140), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,61)(8,62)(9,63)(10,64)(11,126)(12,127)(13,128)(14,129)(15,130)(16,121)(17,122)(18,123)(19,124)(20,125)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,81)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,132,25,150)(2,131,26,149)(3,140,27,148)(4,139,28,147)(5,138,29,146)(6,137,30,145)(7,136,21,144)(8,135,22,143)(9,134,23,142)(10,133,24,141)(11,49,158,32)(12,48,159,31)(13,47,160,40)(14,46,151,39)(15,45,152,38)(16,44,153,37)(17,43,154,36)(18,42,155,35)(19,41,156,34)(20,50,157,33)(51,107,68,99)(52,106,69,98)(53,105,70,97)(54,104,61,96)(55,103,62,95)(56,102,63,94)(57,101,64,93)(58,110,65,92)(59,109,66,91)(60,108,67,100)(71,127,89,119)(72,126,90,118)(73,125,81,117)(74,124,82,116)(75,123,83,115)(76,122,84,114)(77,121,85,113)(78,130,86,112)(79,129,87,111)(80,128,88,120), (1,130,30,117)(2,129,21,116)(3,128,22,115)(4,127,23,114)(5,126,24,113)(6,125,25,112)(7,124,26,111)(8,123,27,120)(9,122,28,119)(10,121,29,118)(11,57,153,69)(12,56,154,68)(13,55,155,67)(14,54,156,66)(15,53,157,65)(16,52,158,64)(17,51,159,63)(18,60,160,62)(19,59,151,61)(20,58,152,70)(31,107,43,94)(32,106,44,93)(33,105,45,92)(34,104,46,91)(35,103,47,100)(36,102,48,99)(37,101,49,98)(38,110,50,97)(39,109,41,96)(40,108,42,95)(71,147,84,134)(72,146,85,133)(73,145,86,132)(74,144,87,131)(75,143,88,140)(76,142,89,139)(77,141,90,138)(78,150,81,137)(79,149,82,136)(80,148,83,135) );
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)], [(1,86,30,73),(2,87,21,74),(3,88,22,75),(4,89,23,76),(5,90,24,77),(6,81,25,78),(7,82,26,79),(8,83,27,80),(9,84,28,71),(10,85,29,72),(11,106,153,93),(12,107,154,94),(13,108,155,95),(14,109,156,96),(15,110,157,97),(16,101,158,98),(17,102,159,99),(18,103,160,100),(19,104,151,91),(20,105,152,92),(31,51,43,63),(32,52,44,64),(33,53,45,65),(34,54,46,66),(35,55,47,67),(36,56,48,68),(37,57,49,69),(38,58,50,70),(39,59,41,61),(40,60,42,62),(111,144,124,131),(112,145,125,132),(113,146,126,133),(114,147,127,134),(115,148,128,135),(116,149,129,136),(117,150,130,137),(118,141,121,138),(119,142,122,139),(120,143,123,140)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,61),(8,62),(9,63),(10,64),(11,126),(12,127),(13,128),(14,129),(15,130),(16,121),(17,122),(18,123),(19,124),(20,125),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,51),(29,52),(30,53),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,81),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,132,25,150),(2,131,26,149),(3,140,27,148),(4,139,28,147),(5,138,29,146),(6,137,30,145),(7,136,21,144),(8,135,22,143),(9,134,23,142),(10,133,24,141),(11,49,158,32),(12,48,159,31),(13,47,160,40),(14,46,151,39),(15,45,152,38),(16,44,153,37),(17,43,154,36),(18,42,155,35),(19,41,156,34),(20,50,157,33),(51,107,68,99),(52,106,69,98),(53,105,70,97),(54,104,61,96),(55,103,62,95),(56,102,63,94),(57,101,64,93),(58,110,65,92),(59,109,66,91),(60,108,67,100),(71,127,89,119),(72,126,90,118),(73,125,81,117),(74,124,82,116),(75,123,83,115),(76,122,84,114),(77,121,85,113),(78,130,86,112),(79,129,87,111),(80,128,88,120)], [(1,130,30,117),(2,129,21,116),(3,128,22,115),(4,127,23,114),(5,126,24,113),(6,125,25,112),(7,124,26,111),(8,123,27,120),(9,122,28,119),(10,121,29,118),(11,57,153,69),(12,56,154,68),(13,55,155,67),(14,54,156,66),(15,53,157,65),(16,52,158,64),(17,51,159,63),(18,60,160,62),(19,59,151,61),(20,58,152,70),(31,107,43,94),(32,106,44,93),(33,105,45,92),(34,104,46,91),(35,103,47,100),(36,102,48,99),(37,101,49,98),(38,110,50,97),(39,109,41,96),(40,108,42,95),(71,147,84,134),(72,146,85,133),(73,145,86,132),(74,144,87,131),(75,143,88,140),(76,142,89,139),(77,141,90,138),(78,150,81,137),(79,149,82,136),(80,148,83,135)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 34 | 0 | 34 | 34 |
| 0 | 0 | 1 | 7 | 7 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 25 | 25 |
| 0 | 0 | 3 | 18 | 0 | 39 |
| 0 | 0 | 35 | 40 | 17 | 40 |
| 0 | 0 | 17 | 0 | 17 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 34 | 40 | 39 | 34 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 2 | 22 | 37 |
| 0 | 0 | 38 | 23 | 28 | 15 |
| 0 | 0 | 6 | 0 | 39 | 6 |
| 0 | 0 | 29 | 39 | 39 | 6 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 28 | 0 | 0 |
| 0 | 0 | 12 | 14 | 0 | 0 |
| 0 | 0 | 18 | 0 | 28 | 18 |
| 0 | 0 | 27 | 13 | 27 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,6,34,1,0,0,34,35,0,7,0,0,0,0,34,7,0,0,0,0,34,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,3,35,17,0,0,1,18,40,0,0,0,25,0,17,17,0,0,25,39,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,0,1,0,0,0,40,0,0,0,0,40,39,1,1,0,0,1,34,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,38,6,29,0,0,2,23,0,39,0,0,22,28,39,39,0,0,37,15,6,6],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,27,12,18,27,0,0,28,14,0,13,0,0,0,0,28,27,0,0,0,0,18,13] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊2D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.702- (1+4) | Dic5.14D4 | C23.D10 | Dic5.Q8 | C2×C10.D4 | C20.48D4 | D4×Dic5 | C23.18D10 | C5×C4⋊D4 | C4⋊D4 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 1 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{70}2_-^{(1+4)} % in TeX
G:=Group("C10.70ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1274);
// by ID
G=gap.SmallGroup(320,1274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=a^5*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^5*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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