direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4.C8, D4.C40, Q8.C40, C40.107D4, M5(2)⋊4C10, M4(2).3C20, (C2×C80)⋊6C2, (C2×C16)⋊2C10, C4.3(C2×C40), (C5×D4).5C8, C8.27(C5×D4), (C5×Q8).5C8, C20.66(C2×C8), C8○D4.2C10, C4○D4.1C20, (C5×M5(2))⋊12C2, C10.42(C22⋊C8), (C2×C40).442C22, (C5×M4(2)).11C4, (C2×C10).26M4(2), C20.160(C22⋊C4), C22.1(C5×M4(2)), (C5×C4○D4).9C4, (C5×C8○D4).5C2, C2.8(C5×C22⋊C8), (C2×C4).42(C2×C20), (C2×C8).96(C2×C10), C4.30(C5×C22⋊C4), (C2×C20).436(C2×C4), SmallGroup(320,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.C8
G = < a,b,c,d | a5=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >
Smallest permutation representation of C5×D4.C8
►On 160 points
Generators in S160
(1 34 159 91 108)(2 35 160 92 109)(3 36 145 93 110)(4 37 146 94 111)(5 38 147 95 112)(6 39 148 96 97)(7 40 149 81 98)(8 41 150 82 99)(9 42 151 83 100)(10 43 152 84 101)(11 44 153 85 102)(12 45 154 86 103)(13 46 155 87 104)(14 47 156 88 105)(15 48 157 89 106)(16 33 158 90 107)(17 127 64 66 132)(18 128 49 67 133)(19 113 50 68 134)(20 114 51 69 135)(21 115 52 70 136)(22 116 53 71 137)(23 117 54 72 138)(24 118 55 73 139)(25 119 56 74 140)(26 120 57 75 141)(27 121 58 76 142)(28 122 59 77 143)(29 123 60 78 144)(30 124 61 79 129)(31 125 62 80 130)(32 126 63 65 131)
(1 20 9 28)(2 21 10 29)(3 22 11 30)(4 23 12 31)(5 24 13 32)(6 25 14 17)(7 26 15 18)(8 27 16 19)(33 113 41 121)(34 114 42 122)(35 115 43 123)(36 116 44 124)(37 117 45 125)(38 118 46 126)(39 119 47 127)(40 120 48 128)(49 149 57 157)(50 150 58 158)(51 151 59 159)(52 152 60 160)(53 153 61 145)(54 154 62 146)(55 155 63 147)(56 156 64 148)(65 95 73 87)(66 96 74 88)(67 81 75 89)(68 82 76 90)(69 83 77 91)(70 84 78 92)(71 85 79 93)(72 86 80 94)(97 140 105 132)(98 141 106 133)(99 142 107 134)(100 143 108 135)(101 144 109 136)(102 129 110 137)(103 130 111 138)(104 131 112 139)
(1 28)(2 10)(3 22)(5 32)(6 14)(7 26)(9 20)(11 30)(13 24)(15 18)(19 27)(23 31)(34 122)(35 43)(36 116)(38 126)(39 47)(40 120)(42 114)(44 124)(46 118)(48 128)(49 157)(50 58)(51 151)(53 145)(54 62)(55 155)(57 149)(59 159)(61 153)(63 147)(65 95)(67 89)(68 76)(69 83)(71 93)(72 80)(73 87)(75 81)(77 91)(79 85)(84 92)(88 96)(97 105)(98 141)(100 135)(101 109)(102 129)(104 139)(106 133)(108 143)(110 137)(112 131)(113 121)(117 125)(130 138)(134 142)(148 156)(152 160)
(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)
G:=sub<Sym(160)| (1,34,159,91,108)(2,35,160,92,109)(3,36,145,93,110)(4,37,146,94,111)(5,38,147,95,112)(6,39,148,96,97)(7,40,149,81,98)(8,41,150,82,99)(9,42,151,83,100)(10,43,152,84,101)(11,44,153,85,102)(12,45,154,86,103)(13,46,155,87,104)(14,47,156,88,105)(15,48,157,89,106)(16,33,158,90,107)(17,127,64,66,132)(18,128,49,67,133)(19,113,50,68,134)(20,114,51,69,135)(21,115,52,70,136)(22,116,53,71,137)(23,117,54,72,138)(24,118,55,73,139)(25,119,56,74,140)(26,120,57,75,141)(27,121,58,76,142)(28,122,59,77,143)(29,123,60,78,144)(30,124,61,79,129)(31,125,62,80,130)(32,126,63,65,131), (1,20,9,28)(2,21,10,29)(3,22,11,30)(4,23,12,31)(5,24,13,32)(6,25,14,17)(7,26,15,18)(8,27,16,19)(33,113,41,121)(34,114,42,122)(35,115,43,123)(36,116,44,124)(37,117,45,125)(38,118,46,126)(39,119,47,127)(40,120,48,128)(49,149,57,157)(50,150,58,158)(51,151,59,159)(52,152,60,160)(53,153,61,145)(54,154,62,146)(55,155,63,147)(56,156,64,148)(65,95,73,87)(66,96,74,88)(67,81,75,89)(68,82,76,90)(69,83,77,91)(70,84,78,92)(71,85,79,93)(72,86,80,94)(97,140,105,132)(98,141,106,133)(99,142,107,134)(100,143,108,135)(101,144,109,136)(102,129,110,137)(103,130,111,138)(104,131,112,139), (1,28)(2,10)(3,22)(5,32)(6,14)(7,26)(9,20)(11,30)(13,24)(15,18)(19,27)(23,31)(34,122)(35,43)(36,116)(38,126)(39,47)(40,120)(42,114)(44,124)(46,118)(48,128)(49,157)(50,58)(51,151)(53,145)(54,62)(55,155)(57,149)(59,159)(61,153)(63,147)(65,95)(67,89)(68,76)(69,83)(71,93)(72,80)(73,87)(75,81)(77,91)(79,85)(84,92)(88,96)(97,105)(98,141)(100,135)(101,109)(102,129)(104,139)(106,133)(108,143)(110,137)(112,131)(113,121)(117,125)(130,138)(134,142)(148,156)(152,160), (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)>;
G:=Group( (1,34,159,91,108)(2,35,160,92,109)(3,36,145,93,110)(4,37,146,94,111)(5,38,147,95,112)(6,39,148,96,97)(7,40,149,81,98)(8,41,150,82,99)(9,42,151,83,100)(10,43,152,84,101)(11,44,153,85,102)(12,45,154,86,103)(13,46,155,87,104)(14,47,156,88,105)(15,48,157,89,106)(16,33,158,90,107)(17,127,64,66,132)(18,128,49,67,133)(19,113,50,68,134)(20,114,51,69,135)(21,115,52,70,136)(22,116,53,71,137)(23,117,54,72,138)(24,118,55,73,139)(25,119,56,74,140)(26,120,57,75,141)(27,121,58,76,142)(28,122,59,77,143)(29,123,60,78,144)(30,124,61,79,129)(31,125,62,80,130)(32,126,63,65,131), (1,20,9,28)(2,21,10,29)(3,22,11,30)(4,23,12,31)(5,24,13,32)(6,25,14,17)(7,26,15,18)(8,27,16,19)(33,113,41,121)(34,114,42,122)(35,115,43,123)(36,116,44,124)(37,117,45,125)(38,118,46,126)(39,119,47,127)(40,120,48,128)(49,149,57,157)(50,150,58,158)(51,151,59,159)(52,152,60,160)(53,153,61,145)(54,154,62,146)(55,155,63,147)(56,156,64,148)(65,95,73,87)(66,96,74,88)(67,81,75,89)(68,82,76,90)(69,83,77,91)(70,84,78,92)(71,85,79,93)(72,86,80,94)(97,140,105,132)(98,141,106,133)(99,142,107,134)(100,143,108,135)(101,144,109,136)(102,129,110,137)(103,130,111,138)(104,131,112,139), (1,28)(2,10)(3,22)(5,32)(6,14)(7,26)(9,20)(11,30)(13,24)(15,18)(19,27)(23,31)(34,122)(35,43)(36,116)(38,126)(39,47)(40,120)(42,114)(44,124)(46,118)(48,128)(49,157)(50,58)(51,151)(53,145)(54,62)(55,155)(57,149)(59,159)(61,153)(63,147)(65,95)(67,89)(68,76)(69,83)(71,93)(72,80)(73,87)(75,81)(77,91)(79,85)(84,92)(88,96)(97,105)(98,141)(100,135)(101,109)(102,129)(104,139)(106,133)(108,143)(110,137)(112,131)(113,121)(117,125)(130,138)(134,142)(148,156)(152,160), (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) );
G=PermutationGroup([[(1,34,159,91,108),(2,35,160,92,109),(3,36,145,93,110),(4,37,146,94,111),(5,38,147,95,112),(6,39,148,96,97),(7,40,149,81,98),(8,41,150,82,99),(9,42,151,83,100),(10,43,152,84,101),(11,44,153,85,102),(12,45,154,86,103),(13,46,155,87,104),(14,47,156,88,105),(15,48,157,89,106),(16,33,158,90,107),(17,127,64,66,132),(18,128,49,67,133),(19,113,50,68,134),(20,114,51,69,135),(21,115,52,70,136),(22,116,53,71,137),(23,117,54,72,138),(24,118,55,73,139),(25,119,56,74,140),(26,120,57,75,141),(27,121,58,76,142),(28,122,59,77,143),(29,123,60,78,144),(30,124,61,79,129),(31,125,62,80,130),(32,126,63,65,131)], [(1,20,9,28),(2,21,10,29),(3,22,11,30),(4,23,12,31),(5,24,13,32),(6,25,14,17),(7,26,15,18),(8,27,16,19),(33,113,41,121),(34,114,42,122),(35,115,43,123),(36,116,44,124),(37,117,45,125),(38,118,46,126),(39,119,47,127),(40,120,48,128),(49,149,57,157),(50,150,58,158),(51,151,59,159),(52,152,60,160),(53,153,61,145),(54,154,62,146),(55,155,63,147),(56,156,64,148),(65,95,73,87),(66,96,74,88),(67,81,75,89),(68,82,76,90),(69,83,77,91),(70,84,78,92),(71,85,79,93),(72,86,80,94),(97,140,105,132),(98,141,106,133),(99,142,107,134),(100,143,108,135),(101,144,109,136),(102,129,110,137),(103,130,111,138),(104,131,112,139)], [(1,28),(2,10),(3,22),(5,32),(6,14),(7,26),(9,20),(11,30),(13,24),(15,18),(19,27),(23,31),(34,122),(35,43),(36,116),(38,126),(39,47),(40,120),(42,114),(44,124),(46,118),(48,128),(49,157),(50,58),(51,151),(53,145),(54,62),(55,155),(57,149),(59,159),(61,153),(63,147),(65,95),(67,89),(68,76),(69,83),(71,93),(72,80),(73,87),(75,81),(77,91),(79,85),(84,92),(88,96),(97,105),(98,141),(100,135),(101,109),(102,129),(104,139),(106,133),(108,143),(110,137),(112,131),(113,121),(117,125),(130,138),(134,142),(148,156),(152,160)], [(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)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 40A | ··· | 40P | 40Q | ··· | 40X | 40Y | ··· | 40AF | 80A | ··· | 80AF | 80AG | ··· | 80AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | C40 | D4 | M4(2) | C5×D4 | D4.C8 | C5×M4(2) | C5×D4.C8 |
| kernel | C5×D4.C8 | C2×C80 | C5×M5(2) | C5×C8○D4 | C5×M4(2) | C5×C4○D4 | D4.C8 | C5×D4 | C5×Q8 | C2×C16 | M5(2) | C8○D4 | M4(2) | C4○D4 | D4 | Q8 | C40 | C2×C10 | C8 | C5 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 | 2 | 2 | 8 | 8 | 8 | 32 |
Matrix representation of C5×D4.C8 ►in GL2(𝔽241) generated by
| 205 | 0 |
| 0 | 205 |
| 8 | 181 |
| 222 | 233 |
| 8 | 181 |
| 230 | 233 |
| 188 | 32 |
| 219 | 68 |
G:=sub<GL(2,GF(241))| [205,0,0,205],[8,222,181,233],[8,230,181,233],[188,219,32,68] >;
C5×D4.C8 in GAP, Magma, Sage, TeX
C_5\times D_4.C_8
% in TeX
G:=Group("C5xD4.C8"); // GroupNames label
G:=SmallGroup(320,155);
// by ID
G=gap.SmallGroup(320,155);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2530,248,3511,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations
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