direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C5⋊2C8, C20⋊4C8, C42.6D5, C10.6C42, C5⋊3(C4×C8), (C4×C20).6C2, C4.18(C4×D5), C20.44(C2×C4), (C2×C20).21C4, C10.15(C2×C8), (C2×C4).87D10, C2.1(C4×Dic5), (C2×C4).7Dic5, C22.6(C2×Dic5), (C2×C20).101C22, C2.1(C2×C5⋊2C8), (C2×C5⋊2C8).13C2, (C2×C10).44(C2×C4), SmallGroup(160,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×C5⋊2C8 |
Generators and relations for C4×C5⋊2C8
G = < a,b,c | a4=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×C5⋊2C8
►Regular action on 160 points
Generators in S160
(1 85 61 69)(2 86 62 70)(3 87 63 71)(4 88 64 72)(5 81 57 65)(6 82 58 66)(7 83 59 67)(8 84 60 68)(9 91 20 136)(10 92 21 129)(11 93 22 130)(12 94 23 131)(13 95 24 132)(14 96 17 133)(15 89 18 134)(16 90 19 135)(25 73 121 149)(26 74 122 150)(27 75 123 151)(28 76 124 152)(29 77 125 145)(30 78 126 146)(31 79 127 147)(32 80 128 148)(33 45 100 141)(34 46 101 142)(35 47 102 143)(36 48 103 144)(37 41 104 137)(38 42 97 138)(39 43 98 139)(40 44 99 140)(49 116 110 158)(50 117 111 159)(51 118 112 160)(52 119 105 153)(53 120 106 154)(54 113 107 155)(55 114 108 156)(56 115 109 157)
(1 147 132 143 113)(2 114 144 133 148)(3 149 134 137 115)(4 116 138 135 150)(5 151 136 139 117)(6 118 140 129 152)(7 145 130 141 119)(8 120 142 131 146)(9 39 111 81 27)(10 28 82 112 40)(11 33 105 83 29)(12 30 84 106 34)(13 35 107 85 31)(14 32 86 108 36)(15 37 109 87 25)(16 26 88 110 38)(17 128 70 55 103)(18 104 56 71 121)(19 122 72 49 97)(20 98 50 65 123)(21 124 66 51 99)(22 100 52 67 125)(23 126 68 53 101)(24 102 54 69 127)(41 157 63 73 89)(42 90 74 64 158)(43 159 57 75 91)(44 92 76 58 160)(45 153 59 77 93)(46 94 78 60 154)(47 155 61 79 95)(48 96 80 62 156)
(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)| (1,85,61,69)(2,86,62,70)(3,87,63,71)(4,88,64,72)(5,81,57,65)(6,82,58,66)(7,83,59,67)(8,84,60,68)(9,91,20,136)(10,92,21,129)(11,93,22,130)(12,94,23,131)(13,95,24,132)(14,96,17,133)(15,89,18,134)(16,90,19,135)(25,73,121,149)(26,74,122,150)(27,75,123,151)(28,76,124,152)(29,77,125,145)(30,78,126,146)(31,79,127,147)(32,80,128,148)(33,45,100,141)(34,46,101,142)(35,47,102,143)(36,48,103,144)(37,41,104,137)(38,42,97,138)(39,43,98,139)(40,44,99,140)(49,116,110,158)(50,117,111,159)(51,118,112,160)(52,119,105,153)(53,120,106,154)(54,113,107,155)(55,114,108,156)(56,115,109,157), (1,147,132,143,113)(2,114,144,133,148)(3,149,134,137,115)(4,116,138,135,150)(5,151,136,139,117)(6,118,140,129,152)(7,145,130,141,119)(8,120,142,131,146)(9,39,111,81,27)(10,28,82,112,40)(11,33,105,83,29)(12,30,84,106,34)(13,35,107,85,31)(14,32,86,108,36)(15,37,109,87,25)(16,26,88,110,38)(17,128,70,55,103)(18,104,56,71,121)(19,122,72,49,97)(20,98,50,65,123)(21,124,66,51,99)(22,100,52,67,125)(23,126,68,53,101)(24,102,54,69,127)(41,157,63,73,89)(42,90,74,64,158)(43,159,57,75,91)(44,92,76,58,160)(45,153,59,77,93)(46,94,78,60,154)(47,155,61,79,95)(48,96,80,62,156), (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( (1,85,61,69)(2,86,62,70)(3,87,63,71)(4,88,64,72)(5,81,57,65)(6,82,58,66)(7,83,59,67)(8,84,60,68)(9,91,20,136)(10,92,21,129)(11,93,22,130)(12,94,23,131)(13,95,24,132)(14,96,17,133)(15,89,18,134)(16,90,19,135)(25,73,121,149)(26,74,122,150)(27,75,123,151)(28,76,124,152)(29,77,125,145)(30,78,126,146)(31,79,127,147)(32,80,128,148)(33,45,100,141)(34,46,101,142)(35,47,102,143)(36,48,103,144)(37,41,104,137)(38,42,97,138)(39,43,98,139)(40,44,99,140)(49,116,110,158)(50,117,111,159)(51,118,112,160)(52,119,105,153)(53,120,106,154)(54,113,107,155)(55,114,108,156)(56,115,109,157), (1,147,132,143,113)(2,114,144,133,148)(3,149,134,137,115)(4,116,138,135,150)(5,151,136,139,117)(6,118,140,129,152)(7,145,130,141,119)(8,120,142,131,146)(9,39,111,81,27)(10,28,82,112,40)(11,33,105,83,29)(12,30,84,106,34)(13,35,107,85,31)(14,32,86,108,36)(15,37,109,87,25)(16,26,88,110,38)(17,128,70,55,103)(18,104,56,71,121)(19,122,72,49,97)(20,98,50,65,123)(21,124,66,51,99)(22,100,52,67,125)(23,126,68,53,101)(24,102,54,69,127)(41,157,63,73,89)(42,90,74,64,158)(43,159,57,75,91)(44,92,76,58,160)(45,153,59,77,93)(46,94,78,60,154)(47,155,61,79,95)(48,96,80,62,156), (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([[(1,85,61,69),(2,86,62,70),(3,87,63,71),(4,88,64,72),(5,81,57,65),(6,82,58,66),(7,83,59,67),(8,84,60,68),(9,91,20,136),(10,92,21,129),(11,93,22,130),(12,94,23,131),(13,95,24,132),(14,96,17,133),(15,89,18,134),(16,90,19,135),(25,73,121,149),(26,74,122,150),(27,75,123,151),(28,76,124,152),(29,77,125,145),(30,78,126,146),(31,79,127,147),(32,80,128,148),(33,45,100,141),(34,46,101,142),(35,47,102,143),(36,48,103,144),(37,41,104,137),(38,42,97,138),(39,43,98,139),(40,44,99,140),(49,116,110,158),(50,117,111,159),(51,118,112,160),(52,119,105,153),(53,120,106,154),(54,113,107,155),(55,114,108,156),(56,115,109,157)], [(1,147,132,143,113),(2,114,144,133,148),(3,149,134,137,115),(4,116,138,135,150),(5,151,136,139,117),(6,118,140,129,152),(7,145,130,141,119),(8,120,142,131,146),(9,39,111,81,27),(10,28,82,112,40),(11,33,105,83,29),(12,30,84,106,34),(13,35,107,85,31),(14,32,86,108,36),(15,37,109,87,25),(16,26,88,110,38),(17,128,70,55,103),(18,104,56,71,121),(19,122,72,49,97),(20,98,50,65,123),(21,124,66,51,99),(22,100,52,67,125),(23,126,68,53,101),(24,102,54,69,127),(41,157,63,73,89),(42,90,74,64,158),(43,159,57,75,91),(44,92,76,58,160),(45,153,59,77,93),(46,94,78,60,154),(47,155,61,79,95),(48,96,80,62,156)], [(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)]])
C4×C5⋊2C8 is a maximal subgroup of
C42.279D10 C40⋊8C8 C20.53D8 C20.39SD16 D20⋊4C8 Dic10⋊4C8 C20.57D8 C20.26Q16 C20⋊C16 C42.4F5 C42.9F5 D5×C4×C8 C42.282D10 D10.7C42 C42.185D10 C42.196D10 Dic10⋊5C8 C42.198D10 D20⋊5C8 C20⋊6M4(2) C42.6Dic5 C20.35C42 C42.187D10 C20⋊7M4(2) C42.210D10 C42.213D10 C42.214D10 C42.215D10 C42.216D10 C20.16D8 C20⋊D8 C20⋊4SD16 C20.17D8 C20.SD16 C20.Q16 C20⋊6SD16 C20.D8 C20⋊3Q16 C20.11Q16
C4×C5⋊2C8 is a maximal quotient of
C40⋊8C8 C40.10C8 (C2×C20)⋊8C8
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | Dic5 | D10 | C5⋊2C8 | C4×D5 |
| kernel | C4×C5⋊2C8 | C2×C5⋊2C8 | C4×C20 | C5⋊2C8 | C2×C20 | C20 | C42 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 2 | 4 | 2 | 16 | 8 |
Matrix representation of C4×C5⋊2C8 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 6 | 40 |
| 0 | 1 | 0 |
| 38 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 6 | 40 |
G:=sub<GL(3,GF(41))| [1,0,0,0,32,0,0,0,32],[1,0,0,0,6,1,0,40,0],[38,0,0,0,1,6,0,0,40] >;
C4×C5⋊2C8 in GAP, Magma, Sage, TeX
C_4\times C_5\rtimes_2C_8
% in TeX
G:=Group("C4xC5:2C8"); // GroupNames label
G:=SmallGroup(160,9);
// by ID
G=gap.SmallGroup(160,9);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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