direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×Q32, C16.C10, C80.2C2, Q16.C10, C10.17D8, C20.38D4, C40.26C22, C4.3(C5×D4), C2.5(C5×D8), C8.4(C2×C10), (C5×Q16).2C2, SmallGroup(160,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q32
G = < a,b,c | a5=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Q32
►Regular action on 160 points
Generators in S160
(1 128 51 27 48)(2 113 52 28 33)(3 114 53 29 34)(4 115 54 30 35)(5 116 55 31 36)(6 117 56 32 37)(7 118 57 17 38)(8 119 58 18 39)(9 120 59 19 40)(10 121 60 20 41)(11 122 61 21 42)(12 123 62 22 43)(13 124 63 23 44)(14 125 64 24 45)(15 126 49 25 46)(16 127 50 26 47)(65 110 91 143 157)(66 111 92 144 158)(67 112 93 129 159)(68 97 94 130 160)(69 98 95 131 145)(70 99 96 132 146)(71 100 81 133 147)(72 101 82 134 148)(73 102 83 135 149)(74 103 84 136 150)(75 104 85 137 151)(76 105 86 138 152)(77 106 87 139 153)(78 107 88 140 154)(79 108 89 141 155)(80 109 90 142 156)
(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)
(1 70 9 78)(2 69 10 77)(3 68 11 76)(4 67 12 75)(5 66 13 74)(6 65 14 73)(7 80 15 72)(8 79 16 71)(17 142 25 134)(18 141 26 133)(19 140 27 132)(20 139 28 131)(21 138 29 130)(22 137 30 129)(23 136 31 144)(24 135 32 143)(33 145 41 153)(34 160 42 152)(35 159 43 151)(36 158 44 150)(37 157 45 149)(38 156 46 148)(39 155 47 147)(40 154 48 146)(49 82 57 90)(50 81 58 89)(51 96 59 88)(52 95 60 87)(53 94 61 86)(54 93 62 85)(55 92 63 84)(56 91 64 83)(97 122 105 114)(98 121 106 113)(99 120 107 128)(100 119 108 127)(101 118 109 126)(102 117 110 125)(103 116 111 124)(104 115 112 123)
G:=sub<Sym(160)| (1,128,51,27,48)(2,113,52,28,33)(3,114,53,29,34)(4,115,54,30,35)(5,116,55,31,36)(6,117,56,32,37)(7,118,57,17,38)(8,119,58,18,39)(9,120,59,19,40)(10,121,60,20,41)(11,122,61,21,42)(12,123,62,22,43)(13,124,63,23,44)(14,125,64,24,45)(15,126,49,25,46)(16,127,50,26,47)(65,110,91,143,157)(66,111,92,144,158)(67,112,93,129,159)(68,97,94,130,160)(69,98,95,131,145)(70,99,96,132,146)(71,100,81,133,147)(72,101,82,134,148)(73,102,83,135,149)(74,103,84,136,150)(75,104,85,137,151)(76,105,86,138,152)(77,106,87,139,153)(78,107,88,140,154)(79,108,89,141,155)(80,109,90,142,156), (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), (1,70,9,78)(2,69,10,77)(3,68,11,76)(4,67,12,75)(5,66,13,74)(6,65,14,73)(7,80,15,72)(8,79,16,71)(17,142,25,134)(18,141,26,133)(19,140,27,132)(20,139,28,131)(21,138,29,130)(22,137,30,129)(23,136,31,144)(24,135,32,143)(33,145,41,153)(34,160,42,152)(35,159,43,151)(36,158,44,150)(37,157,45,149)(38,156,46,148)(39,155,47,147)(40,154,48,146)(49,82,57,90)(50,81,58,89)(51,96,59,88)(52,95,60,87)(53,94,61,86)(54,93,62,85)(55,92,63,84)(56,91,64,83)(97,122,105,114)(98,121,106,113)(99,120,107,128)(100,119,108,127)(101,118,109,126)(102,117,110,125)(103,116,111,124)(104,115,112,123)>;
G:=Group( (1,128,51,27,48)(2,113,52,28,33)(3,114,53,29,34)(4,115,54,30,35)(5,116,55,31,36)(6,117,56,32,37)(7,118,57,17,38)(8,119,58,18,39)(9,120,59,19,40)(10,121,60,20,41)(11,122,61,21,42)(12,123,62,22,43)(13,124,63,23,44)(14,125,64,24,45)(15,126,49,25,46)(16,127,50,26,47)(65,110,91,143,157)(66,111,92,144,158)(67,112,93,129,159)(68,97,94,130,160)(69,98,95,131,145)(70,99,96,132,146)(71,100,81,133,147)(72,101,82,134,148)(73,102,83,135,149)(74,103,84,136,150)(75,104,85,137,151)(76,105,86,138,152)(77,106,87,139,153)(78,107,88,140,154)(79,108,89,141,155)(80,109,90,142,156), (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), (1,70,9,78)(2,69,10,77)(3,68,11,76)(4,67,12,75)(5,66,13,74)(6,65,14,73)(7,80,15,72)(8,79,16,71)(17,142,25,134)(18,141,26,133)(19,140,27,132)(20,139,28,131)(21,138,29,130)(22,137,30,129)(23,136,31,144)(24,135,32,143)(33,145,41,153)(34,160,42,152)(35,159,43,151)(36,158,44,150)(37,157,45,149)(38,156,46,148)(39,155,47,147)(40,154,48,146)(49,82,57,90)(50,81,58,89)(51,96,59,88)(52,95,60,87)(53,94,61,86)(54,93,62,85)(55,92,63,84)(56,91,64,83)(97,122,105,114)(98,121,106,113)(99,120,107,128)(100,119,108,127)(101,118,109,126)(102,117,110,125)(103,116,111,124)(104,115,112,123) );
G=PermutationGroup([[(1,128,51,27,48),(2,113,52,28,33),(3,114,53,29,34),(4,115,54,30,35),(5,116,55,31,36),(6,117,56,32,37),(7,118,57,17,38),(8,119,58,18,39),(9,120,59,19,40),(10,121,60,20,41),(11,122,61,21,42),(12,123,62,22,43),(13,124,63,23,44),(14,125,64,24,45),(15,126,49,25,46),(16,127,50,26,47),(65,110,91,143,157),(66,111,92,144,158),(67,112,93,129,159),(68,97,94,130,160),(69,98,95,131,145),(70,99,96,132,146),(71,100,81,133,147),(72,101,82,134,148),(73,102,83,135,149),(74,103,84,136,150),(75,104,85,137,151),(76,105,86,138,152),(77,106,87,139,153),(78,107,88,140,154),(79,108,89,141,155),(80,109,90,142,156)], [(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)], [(1,70,9,78),(2,69,10,77),(3,68,11,76),(4,67,12,75),(5,66,13,74),(6,65,14,73),(7,80,15,72),(8,79,16,71),(17,142,25,134),(18,141,26,133),(19,140,27,132),(20,139,28,131),(21,138,29,130),(22,137,30,129),(23,136,31,144),(24,135,32,143),(33,145,41,153),(34,160,42,152),(35,159,43,151),(36,158,44,150),(37,157,45,149),(38,156,46,148),(39,155,47,147),(40,154,48,146),(49,82,57,90),(50,81,58,89),(51,96,59,88),(52,95,60,87),(53,94,61,86),(54,93,62,85),(55,92,63,84),(56,91,64,83),(97,122,105,114),(98,121,106,113),(99,120,107,128),(100,119,108,127),(101,118,109,126),(102,117,110,125),(103,116,111,124),(104,115,112,123)]])
C5×Q32 is a maximal subgroup of
C5⋊SD64 C5⋊Q64 Q32⋊D5 D80⋊5C2
55 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H | 80A | ··· | 80P |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D8 | Q32 | C5×D4 | C5×D8 | C5×Q32 |
| kernel | C5×Q32 | C80 | C5×Q16 | Q32 | C16 | Q16 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C5×Q32 ►in GL2(𝔽31) generated by
| 16 | 0 |
| 0 | 16 |
| 2 | 14 |
| 7 | 3 |
| 0 | 23 |
| 4 | 0 |
G:=sub<GL(2,GF(31))| [16,0,0,16],[2,7,14,3],[0,4,23,0] >;
C5×Q32 in GAP, Magma, Sage, TeX
C_5\times Q_{32} % in TeX
G:=Group("C5xQ32"); // GroupNames label
G:=SmallGroup(160,63);
// by ID
G=gap.SmallGroup(160,63);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,480,265,487,1443,729,165,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c|a^5=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export