direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C5×D32, C32⋊1C10, C160⋊3C2, D16⋊1C10, C40.68D4, C20.39D8, C10.15D16, C80.19C22, C8.5(C5×D4), C4.1(C5×D8), (C5×D16)⋊5C2, C2.3(C5×D16), C16.2(C2×C10), SmallGroup(320,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D32
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D32
►On 160 points
Generators in S160
(1 102 132 71 57)(2 103 133 72 58)(3 104 134 73 59)(4 105 135 74 60)(5 106 136 75 61)(6 107 137 76 62)(7 108 138 77 63)(8 109 139 78 64)(9 110 140 79 33)(10 111 141 80 34)(11 112 142 81 35)(12 113 143 82 36)(13 114 144 83 37)(14 115 145 84 38)(15 116 146 85 39)(16 117 147 86 40)(17 118 148 87 41)(18 119 149 88 42)(19 120 150 89 43)(20 121 151 90 44)(21 122 152 91 45)(22 123 153 92 46)(23 124 154 93 47)(24 125 155 94 48)(25 126 156 95 49)(26 127 157 96 50)(27 128 158 65 51)(28 97 159 66 52)(29 98 160 67 53)(30 99 129 68 54)(31 100 130 69 55)(32 101 131 70 56)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(97 106)(98 105)(99 104)(100 103)(101 102)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(129 134)(130 133)(131 132)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)
G:=sub<Sym(160)| (1,102,132,71,57)(2,103,133,72,58)(3,104,134,73,59)(4,105,135,74,60)(5,106,136,75,61)(6,107,137,76,62)(7,108,138,77,63)(8,109,139,78,64)(9,110,140,79,33)(10,111,141,80,34)(11,112,142,81,35)(12,113,143,82,36)(13,114,144,83,37)(14,115,145,84,38)(15,116,146,85,39)(16,117,147,86,40)(17,118,148,87,41)(18,119,149,88,42)(19,120,150,89,43)(20,121,151,90,44)(21,122,152,91,45)(22,123,153,92,46)(23,124,154,93,47)(24,125,155,94,48)(25,126,156,95,49)(26,127,157,96,50)(27,128,158,65,51)(28,97,159,66,52)(29,98,160,67,53)(30,99,129,68,54)(31,100,130,69,55)(32,101,131,70,56), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,134)(130,133)(131,132)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)>;
G:=Group( (1,102,132,71,57)(2,103,133,72,58)(3,104,134,73,59)(4,105,135,74,60)(5,106,136,75,61)(6,107,137,76,62)(7,108,138,77,63)(8,109,139,78,64)(9,110,140,79,33)(10,111,141,80,34)(11,112,142,81,35)(12,113,143,82,36)(13,114,144,83,37)(14,115,145,84,38)(15,116,146,85,39)(16,117,147,86,40)(17,118,148,87,41)(18,119,149,88,42)(19,120,150,89,43)(20,121,151,90,44)(21,122,152,91,45)(22,123,153,92,46)(23,124,154,93,47)(24,125,155,94,48)(25,126,156,95,49)(26,127,157,96,50)(27,128,158,65,51)(28,97,159,66,52)(29,98,160,67,53)(30,99,129,68,54)(31,100,130,69,55)(32,101,131,70,56), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,134)(130,133)(131,132)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148) );
G=PermutationGroup([[(1,102,132,71,57),(2,103,133,72,58),(3,104,134,73,59),(4,105,135,74,60),(5,106,136,75,61),(6,107,137,76,62),(7,108,138,77,63),(8,109,139,78,64),(9,110,140,79,33),(10,111,141,80,34),(11,112,142,81,35),(12,113,143,82,36),(13,114,144,83,37),(14,115,145,84,38),(15,116,146,85,39),(16,117,147,86,40),(17,118,148,87,41),(18,119,149,88,42),(19,120,150,89,43),(20,121,151,90,44),(21,122,152,91,45),(22,123,153,92,46),(23,124,154,93,47),(24,125,155,94,48),(25,126,156,95,49),(26,127,157,96,50),(27,128,158,65,51),(28,97,159,66,52),(29,98,160,67,53),(30,99,129,68,54),(31,100,130,69,55),(32,101,131,70,56)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(97,106),(98,105),(99,104),(100,103),(101,102),(107,128),(108,127),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118),(129,134),(130,133),(131,132),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 32A | ··· | 32H | 40A | ··· | 40H | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | ··· | 40 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 16 | 16 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D8 | D16 | C5×D4 | D32 | C5×D8 | C5×D16 | C5×D32 |
| kernel | C5×D32 | C160 | C5×D16 | D32 | C32 | D16 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C5×D32 ►in GL2(𝔽31) generated by
| 2 | 0 |
| 0 | 2 |
| 0 | 27 |
| 8 | 27 |
| 27 | 29 |
| 23 | 4 |
G:=sub<GL(2,GF(31))| [2,0,0,2],[0,8,27,27],[27,23,29,4] >;
C5×D32 in GAP, Magma, Sage, TeX
C_5\times D_{32} % in TeX
G:=Group("C5xD32"); // GroupNames label
G:=SmallGroup(320,176);
// by ID
G=gap.SmallGroup(320,176);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,309,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export