direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊8C4, C10⋊4(C4×D4), C4⋊C4⋊52D10, (C2×D20)⋊29C4, D20⋊32(C2×C4), C20⋊6(C22×C4), (C2×Dic5)⋊24D4, Dic5⋊10(C2×D4), D10⋊4(C22×C4), C10.34(C23×C4), (C2×C10).47C24, C10.41(C22×D4), C22.131(D4×D5), (C2×C20).579C23, (C4×Dic5)⋊64C22, (C22×D20).17C2, (C22×C4).317D10, D10⋊C4⋊59C22, C22.23(C23×D5), (C2×D20).260C22, C23.326(C22×D5), (C22×C10).396C23, (C22×C20).215C22, C22.34(Q8⋊2D5), (C2×Dic5).372C23, (C22×D5).164C23, (C23×D5).110C22, (C22×Dic5).234C22, C5⋊4(C2×C4×D4), C4⋊2(C2×C4×D5), C2.4(C2×D4×D5), (C2×C4)⋊9(C4×D5), (C10×C4⋊C4)⋊9C2, (C2×C4⋊C4)⋊26D5, (C2×C4×Dic5)⋊5C2, (C2×C20)⋊21(C2×C4), (D5×C22×C4)⋊19C2, (C2×C4×D5)⋊67C22, (C5×C4⋊C4)⋊44C22, C22.73(C2×C4×D5), C2.15(D5×C22×C4), C2.2(C2×Q8⋊2D5), C10.108(C2×C4○D4), (C2×C10).387(C2×D4), (C22×D5)⋊16(C2×C4), (C2×D10⋊C4)⋊32C2, (C2×C4).266(C22×D5), (C2×C10).196(C4○D4), (C2×C10).253(C22×C4), SmallGroup(320,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1518 in 426 conjugacy classes, 175 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×32], C5, C2×C4 [×10], C2×C4 [×30], D4 [×16], C23, C23 [×20], D5 [×8], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×18], C2×D4 [×12], C24 [×2], Dic5 [×4], Dic5 [×2], C20 [×4], C20 [×4], D10 [×8], D10 [×24], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×16], D20 [×16], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×10], C2×C20 [×4], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×C4×D4, C4×Dic5 [×4], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×C4×D5 [×8], C2×C4×D5 [×8], C2×D20 [×12], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], D20⋊8C4 [×8], C2×C4×Dic5, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4 [×2], C22×D20, C2×D20⋊8C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], D4×D5 [×2], Q8⋊2D5 [×2], C23×D5, D20⋊8C4 [×4], D5×C22×C4, C2×D4×D5, C2×Q8⋊2D5, C2×D20⋊8C4
Generators and relations
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >
►On 160 points
Generators in S160
(1 137)(2 138)(3 139)(4 140)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(96 100)(97 99)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 86 151 45)(2 97 152 56)(3 88 153 47)(4 99 154 58)(5 90 155 49)(6 81 156 60)(7 92 157 51)(8 83 158 42)(9 94 159 53)(10 85 160 44)(11 96 141 55)(12 87 142 46)(13 98 143 57)(14 89 144 48)(15 100 145 59)(16 91 146 50)(17 82 147 41)(18 93 148 52)(19 84 149 43)(20 95 150 54)(21 140 112 63)(22 131 113 74)(23 122 114 65)(24 133 115 76)(25 124 116 67)(26 135 117 78)(27 126 118 69)(28 137 119 80)(29 128 120 71)(30 139 101 62)(31 130 102 73)(32 121 103 64)(33 132 104 75)(34 123 105 66)(35 134 106 77)(36 125 107 68)(37 136 108 79)(38 127 109 70)(39 138 110 61)(40 129 111 72)
G:=sub<Sym(160)| (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,58)(22,59)(23,60)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(96,100)(97,99)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,86,151,45)(2,97,152,56)(3,88,153,47)(4,99,154,58)(5,90,155,49)(6,81,156,60)(7,92,157,51)(8,83,158,42)(9,94,159,53)(10,85,160,44)(11,96,141,55)(12,87,142,46)(13,98,143,57)(14,89,144,48)(15,100,145,59)(16,91,146,50)(17,82,147,41)(18,93,148,52)(19,84,149,43)(20,95,150,54)(21,140,112,63)(22,131,113,74)(23,122,114,65)(24,133,115,76)(25,124,116,67)(26,135,117,78)(27,126,118,69)(28,137,119,80)(29,128,120,71)(30,139,101,62)(31,130,102,73)(32,121,103,64)(33,132,104,75)(34,123,105,66)(35,134,106,77)(36,125,107,68)(37,136,108,79)(38,127,109,70)(39,138,110,61)(40,129,111,72)>;
G:=Group( (1,137)(2,138)(3,139)(4,140)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,58)(22,59)(23,60)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(96,100)(97,99)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159), (1,86,151,45)(2,97,152,56)(3,88,153,47)(4,99,154,58)(5,90,155,49)(6,81,156,60)(7,92,157,51)(8,83,158,42)(9,94,159,53)(10,85,160,44)(11,96,141,55)(12,87,142,46)(13,98,143,57)(14,89,144,48)(15,100,145,59)(16,91,146,50)(17,82,147,41)(18,93,148,52)(19,84,149,43)(20,95,150,54)(21,140,112,63)(22,131,113,74)(23,122,114,65)(24,133,115,76)(25,124,116,67)(26,135,117,78)(27,126,118,69)(28,137,119,80)(29,128,120,71)(30,139,101,62)(31,130,102,73)(32,121,103,64)(33,132,104,75)(34,123,105,66)(35,134,106,77)(36,125,107,68)(37,136,108,79)(38,127,109,70)(39,138,110,61)(40,129,111,72) );
G=PermutationGroup([(1,137),(2,138),(3,139),(4,140),(5,121),(6,122),(7,123),(8,124),(9,125),(10,126),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,58),(22,59),(23,60),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,111),(99,112),(100,113)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89),(96,100),(97,99),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)], [(1,86,151,45),(2,97,152,56),(3,88,153,47),(4,99,154,58),(5,90,155,49),(6,81,156,60),(7,92,157,51),(8,83,158,42),(9,94,159,53),(10,85,160,44),(11,96,141,55),(12,87,142,46),(13,98,143,57),(14,89,144,48),(15,100,145,59),(16,91,146,50),(17,82,147,41),(18,93,148,52),(19,84,149,43),(20,95,150,54),(21,140,112,63),(22,131,113,74),(23,122,114,65),(24,133,115,76),(25,124,116,67),(26,135,117,78),(27,126,118,69),(28,137,119,80),(29,128,120,71),(30,139,101,62),(31,130,102,73),(32,121,103,64),(33,132,104,75),(34,123,105,66),(35,134,106,77),(36,125,107,68),(37,136,108,79),(38,127,109,70),(39,138,110,61),(40,129,111,72)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 9 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,9,0],[40,0,0,0,0,0,34,7,0,0,0,40,7,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,40,0] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 4M | ··· | 4T | 4U | 4V | 4W | 4X | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D4×D5 | Q8⋊2D5 |
| kernel | C2×D20⋊8C4 | D20⋊8C4 | C2×C4×Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | C22×D20 | C2×D20 | C2×Dic5 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 4 | 2 | 4 | 8 | 6 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_8C_4 % in TeX
G:=Group("C2xD20:8C4"); // GroupNames label
G:=SmallGroup(320,1175);
// by ID
G=gap.SmallGroup(320,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀