direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊C4, C8⋊3C20, C40⋊11C4, C42.1C10, C10.12C42, C10.12M4(2), (C2×C4).2C20, C2.2(C4×C20), (C4×C20).1C2, (C2×C8).7C10, (C2×C40).17C2, C20.69(C2×C4), C4.11(C2×C20), (C2×C20).16C4, C22.8(C2×C20), C2.1(C5×M4(2)), (C2×C20).134C22, (C2×C10).57(C2×C4), (C2×C4).30(C2×C10), SmallGroup(160,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊C4
G = < a,b,c | a5=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C5×C8⋊C4
►Regular action on 160 points
Generators in S160
(1 90 50 82 42)(2 91 51 83 43)(3 92 52 84 44)(4 93 53 85 45)(5 94 54 86 46)(6 95 55 87 47)(7 96 56 88 48)(8 89 49 81 41)(9 115 139 17 131)(10 116 140 18 132)(11 117 141 19 133)(12 118 142 20 134)(13 119 143 21 135)(14 120 144 22 136)(15 113 137 23 129)(16 114 138 24 130)(25 128 155 33 147)(26 121 156 34 148)(27 122 157 35 149)(28 123 158 36 150)(29 124 159 37 151)(30 125 160 38 152)(31 126 153 39 145)(32 127 154 40 146)(57 77 109 69 100)(58 78 110 70 101)(59 79 111 71 102)(60 80 112 72 103)(61 73 105 65 104)(62 74 106 66 97)(63 75 107 67 98)(64 76 108 68 99)
(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 69 25 115)(2 66 26 120)(3 71 27 117)(4 68 28 114)(5 65 29 119)(6 70 30 116)(7 67 31 113)(8 72 32 118)(9 42 109 147)(10 47 110 152)(11 44 111 149)(12 41 112 146)(13 46 105 151)(14 43 106 148)(15 48 107 145)(16 45 108 150)(17 50 57 155)(18 55 58 160)(19 52 59 157)(20 49 60 154)(21 54 61 159)(22 51 62 156)(23 56 63 153)(24 53 64 158)(33 131 82 77)(34 136 83 74)(35 133 84 79)(36 130 85 76)(37 135 86 73)(38 132 87 78)(39 129 88 75)(40 134 81 80)(89 103 127 142)(90 100 128 139)(91 97 121 144)(92 102 122 141)(93 99 123 138)(94 104 124 143)(95 101 125 140)(96 98 126 137)
G:=sub<Sym(160)| (1,90,50,82,42)(2,91,51,83,43)(3,92,52,84,44)(4,93,53,85,45)(5,94,54,86,46)(6,95,55,87,47)(7,96,56,88,48)(8,89,49,81,41)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,128,155,33,147)(26,121,156,34,148)(27,122,157,35,149)(28,123,158,36,150)(29,124,159,37,151)(30,125,160,38,152)(31,126,153,39,145)(32,127,154,40,146)(57,77,109,69,100)(58,78,110,70,101)(59,79,111,71,102)(60,80,112,72,103)(61,73,105,65,104)(62,74,106,66,97)(63,75,107,67,98)(64,76,108,68,99), (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), (1,69,25,115)(2,66,26,120)(3,71,27,117)(4,68,28,114)(5,65,29,119)(6,70,30,116)(7,67,31,113)(8,72,32,118)(9,42,109,147)(10,47,110,152)(11,44,111,149)(12,41,112,146)(13,46,105,151)(14,43,106,148)(15,48,107,145)(16,45,108,150)(17,50,57,155)(18,55,58,160)(19,52,59,157)(20,49,60,154)(21,54,61,159)(22,51,62,156)(23,56,63,153)(24,53,64,158)(33,131,82,77)(34,136,83,74)(35,133,84,79)(36,130,85,76)(37,135,86,73)(38,132,87,78)(39,129,88,75)(40,134,81,80)(89,103,127,142)(90,100,128,139)(91,97,121,144)(92,102,122,141)(93,99,123,138)(94,104,124,143)(95,101,125,140)(96,98,126,137)>;
G:=Group( (1,90,50,82,42)(2,91,51,83,43)(3,92,52,84,44)(4,93,53,85,45)(5,94,54,86,46)(6,95,55,87,47)(7,96,56,88,48)(8,89,49,81,41)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,128,155,33,147)(26,121,156,34,148)(27,122,157,35,149)(28,123,158,36,150)(29,124,159,37,151)(30,125,160,38,152)(31,126,153,39,145)(32,127,154,40,146)(57,77,109,69,100)(58,78,110,70,101)(59,79,111,71,102)(60,80,112,72,103)(61,73,105,65,104)(62,74,106,66,97)(63,75,107,67,98)(64,76,108,68,99), (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), (1,69,25,115)(2,66,26,120)(3,71,27,117)(4,68,28,114)(5,65,29,119)(6,70,30,116)(7,67,31,113)(8,72,32,118)(9,42,109,147)(10,47,110,152)(11,44,111,149)(12,41,112,146)(13,46,105,151)(14,43,106,148)(15,48,107,145)(16,45,108,150)(17,50,57,155)(18,55,58,160)(19,52,59,157)(20,49,60,154)(21,54,61,159)(22,51,62,156)(23,56,63,153)(24,53,64,158)(33,131,82,77)(34,136,83,74)(35,133,84,79)(36,130,85,76)(37,135,86,73)(38,132,87,78)(39,129,88,75)(40,134,81,80)(89,103,127,142)(90,100,128,139)(91,97,121,144)(92,102,122,141)(93,99,123,138)(94,104,124,143)(95,101,125,140)(96,98,126,137) );
G=PermutationGroup([[(1,90,50,82,42),(2,91,51,83,43),(3,92,52,84,44),(4,93,53,85,45),(5,94,54,86,46),(6,95,55,87,47),(7,96,56,88,48),(8,89,49,81,41),(9,115,139,17,131),(10,116,140,18,132),(11,117,141,19,133),(12,118,142,20,134),(13,119,143,21,135),(14,120,144,22,136),(15,113,137,23,129),(16,114,138,24,130),(25,128,155,33,147),(26,121,156,34,148),(27,122,157,35,149),(28,123,158,36,150),(29,124,159,37,151),(30,125,160,38,152),(31,126,153,39,145),(32,127,154,40,146),(57,77,109,69,100),(58,78,110,70,101),(59,79,111,71,102),(60,80,112,72,103),(61,73,105,65,104),(62,74,106,66,97),(63,75,107,67,98),(64,76,108,68,99)], [(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)], [(1,69,25,115),(2,66,26,120),(3,71,27,117),(4,68,28,114),(5,65,29,119),(6,70,30,116),(7,67,31,113),(8,72,32,118),(9,42,109,147),(10,47,110,152),(11,44,111,149),(12,41,112,146),(13,46,105,151),(14,43,106,148),(15,48,107,145),(16,45,108,150),(17,50,57,155),(18,55,58,160),(19,52,59,157),(20,49,60,154),(21,54,61,159),(22,51,62,156),(23,56,63,153),(24,53,64,158),(33,131,82,77),(34,136,83,74),(35,133,84,79),(36,130,85,76),(37,135,86,73),(38,132,87,78),(39,129,88,75),(40,134,81,80),(89,103,127,142),(90,100,128,139),(91,97,121,144),(92,102,122,141),(93,99,123,138),(94,104,124,143),(95,101,125,140),(96,98,126,137)]])
C5×C8⋊C4 is a maximal subgroup of
C42.D10 C42.2D10 C20.45C42 C40⋊Q8 C8⋊Dic10 C42.14D10 C42.182D10 C8⋊9D20 D10.6C42 D10.7C42 C42.185D10 C42.16D10 D40⋊9C4 C8⋊D20 C42.19D10 C42.20D10 C8.D20 Dic20⋊9C4 D40⋊10C4 M4(2)×C20
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | M4(2) | C5×M4(2) |
| kernel | C5×C8⋊C4 | C4×C20 | C2×C40 | C40 | C2×C20 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 4 | 4 | 8 | 32 | 16 | 4 | 16 |
Matrix representation of C5×C8⋊C4 ►in GL3(𝔽41) generated by
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 21 | 1 |
| 0 | 1 | 20 |
| 32 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
G:=sub<GL(3,GF(41))| [10,0,0,0,1,0,0,0,1],[1,0,0,0,21,1,0,1,20],[32,0,0,0,0,40,0,1,0] >;
C5×C8⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes C_4
% in TeX
G:=Group("C5xC8:C4"); // GroupNames label
G:=SmallGroup(160,47);
// by ID
G=gap.SmallGroup(160,47);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,985,247,117]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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