direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.48D4, C23⋊4Dic10, C24.67D10, (C22×C10)⋊7Q8, (C2×C20).477D4, C20.424(C2×D4), C10⋊4(C22⋊Q8), (C23×C4).10D5, (C23×C20).12C2, C4⋊Dic5⋊63C22, C22⋊3(C2×Dic10), C10.19(C22×Q8), (C2×C10).282C24, (C2×C20).703C23, C10.130(C22×D4), (C22×C4).446D10, (C22×Dic10)⋊12C2, (C2×Dic10)⋊58C22, C22.79(C4○D20), C10.D4⋊43C22, C2.20(C22×Dic10), C23.231(C22×D5), C22.301(C23×D5), (C22×C10).411C23, (C23×C10).104C22, (C22×C20).528C22, (C2×Dic5).148C23, C23.D5.129C22, (C22×Dic5).160C22, C5⋊5(C2×C22⋊Q8), (C2×C10)⋊6(C2×Q8), (C2×C4⋊Dic5)⋊28C2, C10.59(C2×C4○D4), C2.69(C2×C4○D20), C4.120(C2×C5⋊D4), C2.5(C22×C5⋊D4), (C2×C10).571(C2×D4), (C2×C10.D4)⋊17C2, (C2×C4).262(C5⋊D4), (C2×C4).656(C22×D5), C22.100(C2×C5⋊D4), (C2×C23.D5).23C2, (C2×C10).110(C4○D4), SmallGroup(320,1456)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 878 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C5, C2×C4 [×8], C2×C4 [×26], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×8], C24, Dic5 [×8], C20 [×4], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×8], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C2×C10.D4 [×2], C20.48D4 [×8], C2×C4⋊Dic5, C2×C23.D5 [×2], C22×Dic10, C23×C20, C2×C20.48D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, Dic10 [×4], C5⋊D4 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×Dic10 [×6], C4○D20 [×2], C2×C5⋊D4 [×6], C23×D5, C20.48D4 [×4], C22×Dic10, C2×C4○D20, C22×C5⋊D4, C2×C20.48D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >
►On 160 points
Generators in S160
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 141)(100 142)
(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)
(1 158 117 126)(2 157 118 125)(3 156 119 124)(4 155 120 123)(5 154 101 122)(6 153 102 121)(7 152 103 140)(8 151 104 139)(9 150 105 138)(10 149 106 137)(11 148 107 136)(12 147 108 135)(13 146 109 134)(14 145 110 133)(15 144 111 132)(16 143 112 131)(17 142 113 130)(18 141 114 129)(19 160 115 128)(20 159 116 127)(21 76 99 52)(22 75 100 51)(23 74 81 50)(24 73 82 49)(25 72 83 48)(26 71 84 47)(27 70 85 46)(28 69 86 45)(29 68 87 44)(30 67 88 43)(31 66 89 42)(32 65 90 41)(33 64 91 60)(34 63 92 59)(35 62 93 58)(36 61 94 57)(37 80 95 56)(38 79 96 55)(39 78 97 54)(40 77 98 53)
(1 136 11 126)(2 135 12 125)(3 134 13 124)(4 133 14 123)(5 132 15 122)(6 131 16 121)(7 130 17 140)(8 129 18 139)(9 128 19 138)(10 127 20 137)(21 76 31 66)(22 75 32 65)(23 74 33 64)(24 73 34 63)(25 72 35 62)(26 71 36 61)(27 70 37 80)(28 69 38 79)(29 68 39 78)(30 67 40 77)(41 100 51 90)(42 99 52 89)(43 98 53 88)(44 97 54 87)(45 96 55 86)(46 95 56 85)(47 94 57 84)(48 93 58 83)(49 92 59 82)(50 91 60 81)(101 144 111 154)(102 143 112 153)(103 142 113 152)(104 141 114 151)(105 160 115 150)(106 159 116 149)(107 158 117 148)(108 157 118 147)(109 156 119 146)(110 155 120 145)
G:=sub<Sym(160)| (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,141)(100,142), (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), (1,158,117,126)(2,157,118,125)(3,156,119,124)(4,155,120,123)(5,154,101,122)(6,153,102,121)(7,152,103,140)(8,151,104,139)(9,150,105,138)(10,149,106,137)(11,148,107,136)(12,147,108,135)(13,146,109,134)(14,145,110,133)(15,144,111,132)(16,143,112,131)(17,142,113,130)(18,141,114,129)(19,160,115,128)(20,159,116,127)(21,76,99,52)(22,75,100,51)(23,74,81,50)(24,73,82,49)(25,72,83,48)(26,71,84,47)(27,70,85,46)(28,69,86,45)(29,68,87,44)(30,67,88,43)(31,66,89,42)(32,65,90,41)(33,64,91,60)(34,63,92,59)(35,62,93,58)(36,61,94,57)(37,80,95,56)(38,79,96,55)(39,78,97,54)(40,77,98,53), (1,136,11,126)(2,135,12,125)(3,134,13,124)(4,133,14,123)(5,132,15,122)(6,131,16,121)(7,130,17,140)(8,129,18,139)(9,128,19,138)(10,127,20,137)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(41,100,51,90)(42,99,52,89)(43,98,53,88)(44,97,54,87)(45,96,55,86)(46,95,56,85)(47,94,57,84)(48,93,58,83)(49,92,59,82)(50,91,60,81)(101,144,111,154)(102,143,112,153)(103,142,113,152)(104,141,114,151)(105,160,115,150)(106,159,116,149)(107,158,117,148)(108,157,118,147)(109,156,119,146)(110,155,120,145)>;
G:=Group( (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,141)(100,142), (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), (1,158,117,126)(2,157,118,125)(3,156,119,124)(4,155,120,123)(5,154,101,122)(6,153,102,121)(7,152,103,140)(8,151,104,139)(9,150,105,138)(10,149,106,137)(11,148,107,136)(12,147,108,135)(13,146,109,134)(14,145,110,133)(15,144,111,132)(16,143,112,131)(17,142,113,130)(18,141,114,129)(19,160,115,128)(20,159,116,127)(21,76,99,52)(22,75,100,51)(23,74,81,50)(24,73,82,49)(25,72,83,48)(26,71,84,47)(27,70,85,46)(28,69,86,45)(29,68,87,44)(30,67,88,43)(31,66,89,42)(32,65,90,41)(33,64,91,60)(34,63,92,59)(35,62,93,58)(36,61,94,57)(37,80,95,56)(38,79,96,55)(39,78,97,54)(40,77,98,53), (1,136,11,126)(2,135,12,125)(3,134,13,124)(4,133,14,123)(5,132,15,122)(6,131,16,121)(7,130,17,140)(8,129,18,139)(9,128,19,138)(10,127,20,137)(21,76,31,66)(22,75,32,65)(23,74,33,64)(24,73,34,63)(25,72,35,62)(26,71,36,61)(27,70,37,80)(28,69,38,79)(29,68,39,78)(30,67,40,77)(41,100,51,90)(42,99,52,89)(43,98,53,88)(44,97,54,87)(45,96,55,86)(46,95,56,85)(47,94,57,84)(48,93,58,83)(49,92,59,82)(50,91,60,81)(101,144,111,154)(102,143,112,153)(103,142,113,152)(104,141,114,151)(105,160,115,150)(106,159,116,149)(107,158,117,148)(108,157,118,147)(109,156,119,146)(110,155,120,145) );
G=PermutationGroup([(1,79),(2,80),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,101),(60,102),(81,143),(82,144),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(97,159),(98,160),(99,141),(100,142)], [(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)], [(1,158,117,126),(2,157,118,125),(3,156,119,124),(4,155,120,123),(5,154,101,122),(6,153,102,121),(7,152,103,140),(8,151,104,139),(9,150,105,138),(10,149,106,137),(11,148,107,136),(12,147,108,135),(13,146,109,134),(14,145,110,133),(15,144,111,132),(16,143,112,131),(17,142,113,130),(18,141,114,129),(19,160,115,128),(20,159,116,127),(21,76,99,52),(22,75,100,51),(23,74,81,50),(24,73,82,49),(25,72,83,48),(26,71,84,47),(27,70,85,46),(28,69,86,45),(29,68,87,44),(30,67,88,43),(31,66,89,42),(32,65,90,41),(33,64,91,60),(34,63,92,59),(35,62,93,58),(36,61,94,57),(37,80,95,56),(38,79,96,55),(39,78,97,54),(40,77,98,53)], [(1,136,11,126),(2,135,12,125),(3,134,13,124),(4,133,14,123),(5,132,15,122),(6,131,16,121),(7,130,17,140),(8,129,18,139),(9,128,19,138),(10,127,20,137),(21,76,31,66),(22,75,32,65),(23,74,33,64),(24,73,34,63),(25,72,35,62),(26,71,36,61),(27,70,37,80),(28,69,38,79),(29,68,39,78),(30,67,40,77),(41,100,51,90),(42,99,52,89),(43,98,53,88),(44,97,54,87),(45,96,55,86),(46,95,56,85),(47,94,57,84),(48,93,58,83),(49,92,59,82),(50,91,60,81),(101,144,111,154),(102,143,112,153),(103,142,113,152),(104,141,114,151),(105,160,115,150),(106,159,116,149),(107,158,117,148),(108,157,118,147),(109,156,119,146),(110,155,120,145)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 32 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 32 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,4,0,0,0,0,0,31,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0] >;
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Dic10 | C4○D20 |
| kernel | C2×C20.48D4 | C2×C10.D4 | C20.48D4 | C2×C4⋊Dic5 | C2×C23.D5 | C22×Dic10 | C23×C20 | C2×C20 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 12 | 2 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}._{48}D_4 % in TeX
G:=Group("C2xC20.48D4"); // GroupNames label
G:=SmallGroup(320,1456);
// by ID
G=gap.SmallGroup(320,1456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀