direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.D10, C40.8C23, C20.59C24, C23.53D20, M4(2)⋊18D10, Dic20⋊8C22, D20.22C23, Dic10.22C23, (C2×C4).58D20, C4.49(C2×D20), C8.8(C22×D5), (C2×C8).101D10, (C2×C20).204D4, C20.293(C2×D4), C40⋊C2⋊9C22, (C2×M4(2))⋊4D5, C4.56(C23×D5), (C2×Dic20)⋊14C2, C10⋊1(C8.C22), (C10×M4(2))⋊4C2, (C2×C40).69C22, C2.28(C22×D20), C22.74(C2×D20), C10.26(C22×D4), (C2×C20).512C23, C4○D20.50C22, (C22×C4).266D10, (C22×C10).119D4, (C2×Dic10)⋊63C22, (C22×Dic10)⋊18C2, (C2×D20).238C22, (C5×M4(2))⋊20C22, (C22×C20).267C22, C5⋊1(C2×C8.C22), (C2×C40⋊C2)⋊5C2, (C2×C10).63(C2×D4), (C2×C4○D20).23C2, (C2×C4).224(C22×D5), SmallGroup(320,1419)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], Dic5 [×6], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C40 [×4], Dic10 [×6], Dic10 [×7], C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×7], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C2×C8.C22, C40⋊C2 [×8], Dic20 [×8], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×Dic10 [×6], C2×Dic10 [×3], C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C2×C40⋊C2 [×2], C2×Dic20 [×2], C8.D10 [×8], C10×M4(2), C22×Dic10, C2×C4○D20, C2×C8.D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, D20 [×4], C22×D5 [×7], C2×C8.C22, C2×D20 [×6], C23×D5, C8.D10 [×2], C22×D20, C2×C8.D10
Generators and relations
G = < a,b,c,d | a2=b8=1, c10=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 121)(119 122)(120 123)
(1 156 140 33 11 146 130 23)(2 147 121 24 12 157 131 34)(3 158 122 35 13 148 132 25)(4 149 123 26 14 159 133 36)(5 160 124 37 15 150 134 27)(6 151 125 28 16 141 135 38)(7 142 126 39 17 152 136 29)(8 153 127 30 18 143 137 40)(9 144 128 21 19 154 138 31)(10 155 129 32 20 145 139 22)(41 100 117 77 51 90 107 67)(42 91 118 68 52 81 108 78)(43 82 119 79 53 92 109 69)(44 93 120 70 54 83 110 80)(45 84 101 61 55 94 111 71)(46 95 102 72 56 85 112 62)(47 86 103 63 57 96 113 73)(48 97 104 74 58 87 114 64)(49 88 105 65 59 98 115 75)(50 99 106 76 60 89 116 66)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 157 31 147)(22 146 32 156)(23 155 33 145)(24 144 34 154)(25 153 35 143)(26 142 36 152)(27 151 37 141)(28 160 38 150)(29 149 39 159)(30 158 40 148)(41 50 51 60)(42 59 52 49)(43 48 53 58)(44 57 54 47)(45 46 55 56)(61 85 71 95)(62 94 72 84)(63 83 73 93)(64 92 74 82)(65 81 75 91)(66 90 76 100)(67 99 77 89)(68 88 78 98)(69 97 79 87)(70 86 80 96)(101 112 111 102)(103 110 113 120)(104 119 114 109)(105 108 115 118)(106 117 116 107)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)
G:=sub<Sym(160)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,156,140,33,11,146,130,23)(2,147,121,24,12,157,131,34)(3,158,122,35,13,148,132,25)(4,149,123,26,14,159,133,36)(5,160,124,37,15,150,134,27)(6,151,125,28,16,141,135,38)(7,142,126,39,17,152,136,29)(8,153,127,30,18,143,137,40)(9,144,128,21,19,154,138,31)(10,155,129,32,20,145,139,22)(41,100,117,77,51,90,107,67)(42,91,118,68,52,81,108,78)(43,82,119,79,53,92,109,69)(44,93,120,70,54,83,110,80)(45,84,101,61,55,94,111,71)(46,95,102,72,56,85,112,62)(47,86,103,63,57,96,113,73)(48,97,104,74,58,87,114,64)(49,88,105,65,59,98,115,75)(50,99,106,76,60,89,116,66), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,157,31,147)(22,146,32,156)(23,155,33,145)(24,144,34,154)(25,153,35,143)(26,142,36,152)(27,151,37,141)(28,160,38,150)(29,149,39,159)(30,158,40,148)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,85,71,95)(62,94,72,84)(63,83,73,93)(64,92,74,82)(65,81,75,91)(66,90,76,100)(67,99,77,89)(68,88,78,98)(69,97,79,87)(70,86,80,96)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,121)(119,122)(120,123), (1,156,140,33,11,146,130,23)(2,147,121,24,12,157,131,34)(3,158,122,35,13,148,132,25)(4,149,123,26,14,159,133,36)(5,160,124,37,15,150,134,27)(6,151,125,28,16,141,135,38)(7,142,126,39,17,152,136,29)(8,153,127,30,18,143,137,40)(9,144,128,21,19,154,138,31)(10,155,129,32,20,145,139,22)(41,100,117,77,51,90,107,67)(42,91,118,68,52,81,108,78)(43,82,119,79,53,92,109,69)(44,93,120,70,54,83,110,80)(45,84,101,61,55,94,111,71)(46,95,102,72,56,85,112,62)(47,86,103,63,57,96,113,73)(48,97,104,74,58,87,114,64)(49,88,105,65,59,98,115,75)(50,99,106,76,60,89,116,66), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,157,31,147)(22,146,32,156)(23,155,33,145)(24,144,34,154)(25,153,35,143)(26,142,36,152)(27,151,37,141)(28,160,38,150)(29,149,39,159)(30,158,40,148)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,85,71,95)(62,94,72,84)(63,83,73,93)(64,92,74,82)(65,81,75,91)(66,90,76,100)(67,99,77,89)(68,88,78,98)(69,97,79,87)(70,86,80,96)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130) );
G=PermutationGroup([(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64),(81,157),(82,158),(83,159),(84,160),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,121),(119,122),(120,123)], [(1,156,140,33,11,146,130,23),(2,147,121,24,12,157,131,34),(3,158,122,35,13,148,132,25),(4,149,123,26,14,159,133,36),(5,160,124,37,15,150,134,27),(6,151,125,28,16,141,135,38),(7,142,126,39,17,152,136,29),(8,153,127,30,18,143,137,40),(9,144,128,21,19,154,138,31),(10,155,129,32,20,145,139,22),(41,100,117,77,51,90,107,67),(42,91,118,68,52,81,108,78),(43,82,119,79,53,92,109,69),(44,93,120,70,54,83,110,80),(45,84,101,61,55,94,111,71),(46,95,102,72,56,85,112,62),(47,86,103,63,57,96,113,73),(48,97,104,74,58,87,114,64),(49,88,105,65,59,98,115,75),(50,99,106,76,60,89,116,66)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,157,31,147),(22,146,32,156),(23,155,33,145),(24,144,34,154),(25,153,35,143),(26,142,36,152),(27,151,37,141),(28,160,38,150),(29,149,39,159),(30,158,40,148),(41,50,51,60),(42,59,52,49),(43,48,53,58),(44,57,54,47),(45,46,55,56),(61,85,71,95),(62,94,72,84),(63,83,73,93),(64,92,74,82),(65,81,75,91),(66,90,76,100),(67,99,77,89),(68,88,78,98),(69,97,79,87),(70,86,80,96),(101,112,111,102),(103,110,113,120),(104,119,114,109),(105,108,115,118),(106,117,116,107),(121,128,131,138),(122,137,132,127),(123,126,133,136),(124,135,134,125),(129,140,139,130)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 34 | 8 |
| 0 | 0 | 6 | 1 | 33 | 27 |
| 0 | 0 | 33 | 7 | 35 | 6 |
| 0 | 0 | 34 | 32 | 35 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 35 | 32 |
| 0 | 0 | 1 | 0 | 27 | 6 |
| 0 | 0 | 35 | 32 | 28 | 2 |
| 0 | 0 | 27 | 6 | 39 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,1,0,0,0,0,0,0,1,0,0],[0,6,0,0,0,0,34,6,0,0,0,0,0,0,6,6,33,34,0,0,35,1,7,32,0,0,34,33,35,35,0,0,8,27,6,40],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,0,1,35,27,0,0,1,0,32,6,0,0,35,27,28,39,0,0,32,6,2,13] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | C8.C22 | C8.D10 |
| kernel | C2×C8.D10 | C2×C40⋊C2 | C2×Dic20 | C8.D10 | C10×M4(2) | C22×Dic10 | C2×C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8.D_{10} % in TeX
G:=Group("C2xC8.D10"); // GroupNames label
G:=SmallGroup(320,1419);
// by ID
G=gap.SmallGroup(320,1419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^10=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀