direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C4○D8, C20.81C24, C40.79C23, D8⋊6(C2×C10), (C2×D8)⋊13C10, (C10×D8)⋊27C2, (C22×C8)⋊8C10, Q16⋊6(C2×C10), C4.84(D4×C10), (C22×C40)⋊22C2, (C2×C40)⋊51C22, (C10×Q16)⋊27C2, (C2×Q16)⋊13C10, SD16⋊5(C2×C10), (C2×C20).539D4, C20.328(C2×D4), (C5×D8)⋊23C22, C4.4(C23×C10), C22.4(D4×C10), C23.28(C5×D4), (C2×SD16)⋊16C10, (C10×SD16)⋊33C2, C8.12(C22×C10), (C5×Q16)⋊20C22, (C5×D4).35C23, D4.2(C22×C10), Q8.2(C22×C10), (C5×Q8).36C23, (C2×C20).974C23, (C5×SD16)⋊22C22, (C22×C10).132D4, C10.202(C22×D4), (D4×C10).328C22, (Q8×C10).281C22, (C22×C20).604C22, C2.26(D4×C2×C10), (C2×C8)⋊13(C2×C10), C4○D4⋊3(C2×C10), (C2×C4○D4)⋊10C10, (C10×C4○D4)⋊26C2, (C2×C4).148(C5×D4), (C2×D4).74(C2×C10), (C2×C10).689(C2×D4), (C5×C4○D4)⋊23C22, (C2×Q8).69(C2×C10), (C22×C4).131(C2×C10), (C2×C4).144(C22×C10), SmallGroup(320,1574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4○D8
G = < a,b,c,d | a10=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 402 in 266 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C2×C4○D8, C2×C40, C2×C40, C5×D8, C5×SD16, C5×Q16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C22×C40, C10×D8, C10×SD16, C10×Q16, C5×C4○D8, C10×C4○D4, C10×C4○D8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C4○D8, C22×D4, C5×D4, C22×C10, C2×C4○D8, D4×C10, C23×C10, C5×C4○D8, D4×C2×C10, C10×C4○D8
►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)
(1 108 67 79)(2 109 68 80)(3 110 69 71)(4 101 70 72)(5 102 61 73)(6 103 62 74)(7 104 63 75)(8 105 64 76)(9 106 65 77)(10 107 66 78)(11 114 159 143)(12 115 160 144)(13 116 151 145)(14 117 152 146)(15 118 153 147)(16 119 154 148)(17 120 155 149)(18 111 156 150)(19 112 157 141)(20 113 158 142)(21 132 40 123)(22 133 31 124)(23 134 32 125)(24 135 33 126)(25 136 34 127)(26 137 35 128)(27 138 36 129)(28 139 37 130)(29 140 38 121)(30 131 39 122)(41 86 60 95)(42 87 51 96)(43 88 52 97)(44 89 53 98)(45 90 54 99)(46 81 55 100)(47 82 56 91)(48 83 57 92)(49 84 58 93)(50 85 59 94)
(1 143 50 139 67 114 59 130)(2 144 41 140 68 115 60 121)(3 145 42 131 69 116 51 122)(4 146 43 132 70 117 52 123)(5 147 44 133 61 118 53 124)(6 148 45 134 62 119 54 125)(7 149 46 135 63 120 55 126)(8 150 47 136 64 111 56 127)(9 141 48 137 65 112 57 128)(10 142 49 138 66 113 58 129)(11 85 37 79 159 94 28 108)(12 86 38 80 160 95 29 109)(13 87 39 71 151 96 30 110)(14 88 40 72 152 97 21 101)(15 89 31 73 153 98 22 102)(16 90 32 74 154 99 23 103)(17 81 33 75 155 100 24 104)(18 82 34 76 156 91 25 105)(19 83 35 77 157 92 26 106)(20 84 36 78 158 93 27 107)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 31)(21 157)(22 158)(23 159)(24 160)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 101)(78 102)(79 103)(80 104)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(131 150)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
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), (1,108,67,79)(2,109,68,80)(3,110,69,71)(4,101,70,72)(5,102,61,73)(6,103,62,74)(7,104,63,75)(8,105,64,76)(9,106,65,77)(10,107,66,78)(11,114,159,143)(12,115,160,144)(13,116,151,145)(14,117,152,146)(15,118,153,147)(16,119,154,148)(17,120,155,149)(18,111,156,150)(19,112,157,141)(20,113,158,142)(21,132,40,123)(22,133,31,124)(23,134,32,125)(24,135,33,126)(25,136,34,127)(26,137,35,128)(27,138,36,129)(28,139,37,130)(29,140,38,121)(30,131,39,122)(41,86,60,95)(42,87,51,96)(43,88,52,97)(44,89,53,98)(45,90,54,99)(46,81,55,100)(47,82,56,91)(48,83,57,92)(49,84,58,93)(50,85,59,94), (1,143,50,139,67,114,59,130)(2,144,41,140,68,115,60,121)(3,145,42,131,69,116,51,122)(4,146,43,132,70,117,52,123)(5,147,44,133,61,118,53,124)(6,148,45,134,62,119,54,125)(7,149,46,135,63,120,55,126)(8,150,47,136,64,111,56,127)(9,141,48,137,65,112,57,128)(10,142,49,138,66,113,58,129)(11,85,37,79,159,94,28,108)(12,86,38,80,160,95,29,109)(13,87,39,71,151,96,30,110)(14,88,40,72,152,97,21,101)(15,89,31,73,153,98,22,102)(16,90,32,74,154,99,23,103)(17,81,33,75,155,100,24,104)(18,82,34,76,156,91,25,105)(19,83,35,77,157,92,26,106)(20,84,36,78,158,93,27,107), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,31)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,101)(78,102)(79,103)(80,104)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149)>;
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), (1,108,67,79)(2,109,68,80)(3,110,69,71)(4,101,70,72)(5,102,61,73)(6,103,62,74)(7,104,63,75)(8,105,64,76)(9,106,65,77)(10,107,66,78)(11,114,159,143)(12,115,160,144)(13,116,151,145)(14,117,152,146)(15,118,153,147)(16,119,154,148)(17,120,155,149)(18,111,156,150)(19,112,157,141)(20,113,158,142)(21,132,40,123)(22,133,31,124)(23,134,32,125)(24,135,33,126)(25,136,34,127)(26,137,35,128)(27,138,36,129)(28,139,37,130)(29,140,38,121)(30,131,39,122)(41,86,60,95)(42,87,51,96)(43,88,52,97)(44,89,53,98)(45,90,54,99)(46,81,55,100)(47,82,56,91)(48,83,57,92)(49,84,58,93)(50,85,59,94), (1,143,50,139,67,114,59,130)(2,144,41,140,68,115,60,121)(3,145,42,131,69,116,51,122)(4,146,43,132,70,117,52,123)(5,147,44,133,61,118,53,124)(6,148,45,134,62,119,54,125)(7,149,46,135,63,120,55,126)(8,150,47,136,64,111,56,127)(9,141,48,137,65,112,57,128)(10,142,49,138,66,113,58,129)(11,85,37,79,159,94,28,108)(12,86,38,80,160,95,29,109)(13,87,39,71,151,96,30,110)(14,88,40,72,152,97,21,101)(15,89,31,73,153,98,22,102)(16,90,32,74,154,99,23,103)(17,81,33,75,155,100,24,104)(18,82,34,76,156,91,25,105)(19,83,35,77,157,92,26,106)(20,84,36,78,158,93,27,107), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,31)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,101)(78,102)(79,103)(80,104)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,121)(131,150)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149) );
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)], [(1,108,67,79),(2,109,68,80),(3,110,69,71),(4,101,70,72),(5,102,61,73),(6,103,62,74),(7,104,63,75),(8,105,64,76),(9,106,65,77),(10,107,66,78),(11,114,159,143),(12,115,160,144),(13,116,151,145),(14,117,152,146),(15,118,153,147),(16,119,154,148),(17,120,155,149),(18,111,156,150),(19,112,157,141),(20,113,158,142),(21,132,40,123),(22,133,31,124),(23,134,32,125),(24,135,33,126),(25,136,34,127),(26,137,35,128),(27,138,36,129),(28,139,37,130),(29,140,38,121),(30,131,39,122),(41,86,60,95),(42,87,51,96),(43,88,52,97),(44,89,53,98),(45,90,54,99),(46,81,55,100),(47,82,56,91),(48,83,57,92),(49,84,58,93),(50,85,59,94)], [(1,143,50,139,67,114,59,130),(2,144,41,140,68,115,60,121),(3,145,42,131,69,116,51,122),(4,146,43,132,70,117,52,123),(5,147,44,133,61,118,53,124),(6,148,45,134,62,119,54,125),(7,149,46,135,63,120,55,126),(8,150,47,136,64,111,56,127),(9,141,48,137,65,112,57,128),(10,142,49,138,66,113,58,129),(11,85,37,79,159,94,28,108),(12,86,38,80,160,95,29,109),(13,87,39,71,151,96,30,110),(14,88,40,72,152,97,21,101),(15,89,31,73,153,98,22,102),(16,90,32,74,154,99,23,103),(17,81,33,75,155,100,24,104),(18,82,34,76,156,91,25,105),(19,83,35,77,157,92,26,106),(20,84,36,78,158,93,27,107)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,61),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,31),(21,157),(22,158),(23,159),(24,160),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,101),(78,102),(79,103),(80,104),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,121),(131,150),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 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 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D8 |
| kernel | C10×C4○D8 | C22×C40 | C10×D8 | C10×SD16 | C10×Q16 | C5×C4○D8 | C10×C4○D4 | C2×C4○D8 | C22×C8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | C2×C4○D4 | C2×C20 | C22×C10 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 4 | 4 | 4 | 8 | 4 | 32 | 8 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of C10×C4○D8 ►in GL4(𝔽41) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 29 | 29 |
| 40 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [23,0,0,0,0,23,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[1,40,0,0,2,40,0,0,0,0,29,29,0,0,12,29],[40,1,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;
C10×C4○D8 in GAP, Magma, Sage, TeX
C_{10}\times C_4\circ D_8 % in TeX
G:=Group("C10xC4oD8"); // GroupNames label
G:=SmallGroup(320,1574);
// by ID
G=gap.SmallGroup(320,1574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,856,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations