direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4⋊D4, C4⋊3(D4×C10), C20⋊17(C2×D4), (C2×C20)⋊40D4, C23⋊4(C5×D4), (C23×C4)⋊5C10, C22⋊1(D4×C10), (C23×C20)⋊14C2, (C22×D4)⋊4C10, (C22×C10)⋊14D4, (D4×C10)⋊61C22, C24.12(C2×C10), (C2×C10).342C24, (C2×C20).655C23, (C22×C20)⋊65C22, C10.181(C22×D4), C22.16(C23×C10), C23.69(C22×C10), (C23×C10).12C22, (C22×C10).257C23, C2.5(D4×C2×C10), C4⋊C4⋊9(C2×C10), (D4×C2×C10)⋊19C2, (C2×C4)⋊10(C5×D4), (C10×C4⋊C4)⋊41C2, (C2×C4⋊C4)⋊14C10, (C2×D4)⋊9(C2×C10), (C2×C10)⋊10(C2×D4), C2.5(C10×C4○D4), (C5×C4⋊C4)⋊65C22, (C2×C22⋊C4)⋊9C10, C22⋊C4⋊11(C2×C10), (C10×C22⋊C4)⋊29C2, (C22×C4)⋊18(C2×C10), C10.224(C2×C4○D4), C22.29(C5×C4○D4), (C5×C22⋊C4)⋊65C22, (C2×C4).11(C22×C10), (C2×C10).229(C4○D4), SmallGroup(320,1524)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 706 in 426 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], C5, C2×C4 [×12], C2×C4 [×14], D4 [×24], C23, C23 [×10], C23 [×16], C10 [×3], C10 [×4], C10 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4 [×12], C2×D4 [×12], C24, C24 [×2], C20 [×4], C20 [×6], C2×C10, C2×C10 [×10], C2×C10 [×32], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C2×C20 [×12], C2×C20 [×14], C5×D4 [×24], C22×C10, C22×C10 [×10], C22×C10 [×16], C2×C4⋊D4, C5×C22⋊C4 [×8], C5×C4⋊C4 [×4], C22×C20 [×2], C22×C20 [×6], C22×C20 [×4], D4×C10 [×12], D4×C10 [×12], C23×C10, C23×C10 [×2], C10×C22⋊C4 [×2], C10×C4⋊C4, C5×C4⋊D4 [×8], C23×C20, D4×C2×C10, D4×C2×C10 [×2], C10×C4⋊D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×8], C23 [×15], C10 [×15], C2×D4 [×12], C4○D4 [×2], C24, C2×C10 [×35], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C5×D4 [×8], C22×C10 [×15], C2×C4⋊D4, D4×C10 [×12], C5×C4○D4 [×2], C23×C10, C5×C4⋊D4 [×4], D4×C2×C10 [×2], C10×C4○D4, C10×C4⋊D4
Generators and relations
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
►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)
(1 119 56 147)(2 120 57 148)(3 111 58 149)(4 112 59 150)(5 113 60 141)(6 114 51 142)(7 115 52 143)(8 116 53 144)(9 117 54 145)(10 118 55 146)(11 78 34 106)(12 79 35 107)(13 80 36 108)(14 71 37 109)(15 72 38 110)(16 73 39 101)(17 74 40 102)(18 75 31 103)(19 76 32 104)(20 77 33 105)(21 82 153 100)(22 83 154 91)(23 84 155 92)(24 85 156 93)(25 86 157 94)(26 87 158 95)(27 88 159 96)(28 89 160 97)(29 90 151 98)(30 81 152 99)(41 126 65 134)(42 127 66 135)(43 128 67 136)(44 129 68 137)(45 130 69 138)(46 121 70 139)(47 122 61 140)(48 123 62 131)(49 124 63 132)(50 125 64 133)
(1 88 48 74)(2 89 49 75)(3 90 50 76)(4 81 41 77)(5 82 42 78)(6 83 43 79)(7 84 44 80)(8 85 45 71)(9 86 46 72)(10 87 47 73)(11 113 21 127)(12 114 22 128)(13 115 23 129)(14 116 24 130)(15 117 25 121)(16 118 26 122)(17 119 27 123)(18 120 28 124)(19 111 29 125)(20 112 30 126)(31 148 160 132)(32 149 151 133)(33 150 152 134)(34 141 153 135)(35 142 154 136)(36 143 155 137)(37 144 156 138)(38 145 157 139)(39 146 158 140)(40 147 159 131)(51 91 67 107)(52 92 68 108)(53 93 69 109)(54 94 70 110)(55 95 61 101)(56 96 62 102)(57 97 63 103)(58 98 64 104)(59 99 65 105)(60 100 66 106)
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 101)(79 102)(80 103)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 91)(89 92)(90 93)(111 130)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 141)(151 156)(152 157)(153 158)(154 159)(155 160)
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), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,126,65,134)(42,127,66,135)(43,128,67,136)(44,129,68,137)(45,130,69,138)(46,121,70,139)(47,122,61,140)(48,123,62,131)(49,124,63,132)(50,125,64,133), (1,88,48,74)(2,89,49,75)(3,90,50,76)(4,81,41,77)(5,82,42,78)(6,83,43,79)(7,84,44,80)(8,85,45,71)(9,86,46,72)(10,87,47,73)(11,113,21,127)(12,114,22,128)(13,115,23,129)(14,116,24,130)(15,117,25,121)(16,118,26,122)(17,119,27,123)(18,120,28,124)(19,111,29,125)(20,112,30,126)(31,148,160,132)(32,149,151,133)(33,150,152,134)(34,141,153,135)(35,142,154,136)(36,143,155,137)(37,144,156,138)(38,145,157,139)(39,146,158,140)(40,147,159,131)(51,91,67,107)(52,92,68,108)(53,93,69,109)(54,94,70,110)(55,95,61,101)(56,96,62,102)(57,97,63,103)(58,98,64,104)(59,99,65,105)(60,100,66,106), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)(151,156)(152,157)(153,158)(154,159)(155,160)>;
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), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,126,65,134)(42,127,66,135)(43,128,67,136)(44,129,68,137)(45,130,69,138)(46,121,70,139)(47,122,61,140)(48,123,62,131)(49,124,63,132)(50,125,64,133), (1,88,48,74)(2,89,49,75)(3,90,50,76)(4,81,41,77)(5,82,42,78)(6,83,43,79)(7,84,44,80)(8,85,45,71)(9,86,46,72)(10,87,47,73)(11,113,21,127)(12,114,22,128)(13,115,23,129)(14,116,24,130)(15,117,25,121)(16,118,26,122)(17,119,27,123)(18,120,28,124)(19,111,29,125)(20,112,30,126)(31,148,160,132)(32,149,151,133)(33,150,152,134)(34,141,153,135)(35,142,154,136)(36,143,155,137)(37,144,156,138)(38,145,157,139)(39,146,158,140)(40,147,159,131)(51,91,67,107)(52,92,68,108)(53,93,69,109)(54,94,70,110)(55,95,61,101)(56,96,62,102)(57,97,63,103)(58,98,64,104)(59,99,65,105)(60,100,66,106), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)(151,156)(152,157)(153,158)(154,159)(155,160) );
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)], [(1,119,56,147),(2,120,57,148),(3,111,58,149),(4,112,59,150),(5,113,60,141),(6,114,51,142),(7,115,52,143),(8,116,53,144),(9,117,54,145),(10,118,55,146),(11,78,34,106),(12,79,35,107),(13,80,36,108),(14,71,37,109),(15,72,38,110),(16,73,39,101),(17,74,40,102),(18,75,31,103),(19,76,32,104),(20,77,33,105),(21,82,153,100),(22,83,154,91),(23,84,155,92),(24,85,156,93),(25,86,157,94),(26,87,158,95),(27,88,159,96),(28,89,160,97),(29,90,151,98),(30,81,152,99),(41,126,65,134),(42,127,66,135),(43,128,67,136),(44,129,68,137),(45,130,69,138),(46,121,70,139),(47,122,61,140),(48,123,62,131),(49,124,63,132),(50,125,64,133)], [(1,88,48,74),(2,89,49,75),(3,90,50,76),(4,81,41,77),(5,82,42,78),(6,83,43,79),(7,84,44,80),(8,85,45,71),(9,86,46,72),(10,87,47,73),(11,113,21,127),(12,114,22,128),(13,115,23,129),(14,116,24,130),(15,117,25,121),(16,118,26,122),(17,119,27,123),(18,120,28,124),(19,111,29,125),(20,112,30,126),(31,148,160,132),(32,149,151,133),(33,150,152,134),(34,141,153,135),(35,142,154,136),(36,143,155,137),(37,144,156,138),(38,145,157,139),(39,146,158,140),(40,147,159,131),(51,91,67,107),(52,92,68,108),(53,93,69,109),(54,94,70,110),(55,95,61,101),(56,96,62,102),(57,97,63,103),(58,98,64,104),(59,99,65,105),(60,100,66,106)], [(1,67),(2,68),(3,69),(4,70),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,101),(79,102),(80,103),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,91),(89,92),(90,93),(111,130),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,141),(151,156),(152,157),(153,158),(154,159),(155,160)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
G:=sub<GL(6,GF(41))| [23,0,0,0,0,0,0,23,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10BH | 20A | ··· | 20AF | 20AG | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×C4○D4 |
| kernel | C10×C4⋊D4 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C4⋊D4 | C23×C20 | D4×C2×C10 | C2×C4⋊D4 | C2×C22⋊C4 | C2×C4⋊C4 | C4⋊D4 | C23×C4 | C22×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 8 | 1 | 3 | 4 | 8 | 4 | 32 | 4 | 12 | 4 | 4 | 4 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4\rtimes D_4 % in TeX
G:=Group("C10xC4:D4"); // GroupNames label
G:=SmallGroup(320,1524);
// by ID
G=gap.SmallGroup(320,1524);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀