metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q32⋊2D5, C16.3D10, D10.17D8, Q16.4D10, C40.21C23, C80.10C22, Dic5.19D8, D40.4C22, Dic20.6C22, C4.9(D4×D5), (C5×Q32)⋊4C2, (D5×Q16)⋊5C2, C5⋊Q32⋊4C2, C16⋊D5⋊4C2, C80⋊C2⋊4C2, C2.24(D5×D8), C5⋊2C8.5D4, (C4×D5).10D4, C20.15(C2×D4), C10.40(C2×D8), C5⋊SD32⋊3C2, C5⋊3(Q32⋊C2), (C8×D5).6C22, C8.27(C22×D5), Q8.D10.1C2, C5⋊2C16.2C22, (C5×Q16).5C22, SmallGroup(320,545)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q32⋊D5
G = < a,b,c,d | a16=c5=d2=1, b2=a8, bab-1=a-1, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 406 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, M5(2), SD32, Q32, Q32, C2×Q16, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C5×Q8, Q32⋊C2, C5⋊2C16, C80, C8×D5, D40, Dic20, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, C80⋊C2, C16⋊D5, C5⋊SD32, C5⋊Q32, C5×Q32, D5×Q16, Q8.D10, Q32⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, Q32⋊C2, D4×D5, D5×D8, Q32⋊D5
►On 160 points
Generators in S160
(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 80 9 72)(2 79 10 71)(3 78 11 70)(4 77 12 69)(5 76 13 68)(6 75 14 67)(7 74 15 66)(8 73 16 65)(17 39 25 47)(18 38 26 46)(19 37 27 45)(20 36 28 44)(21 35 29 43)(22 34 30 42)(23 33 31 41)(24 48 32 40)(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)(97 140 105 132)(98 139 106 131)(99 138 107 130)(100 137 108 129)(101 136 109 144)(102 135 110 143)(103 134 111 142)(104 133 112 141)(113 149 121 157)(114 148 122 156)(115 147 123 155)(116 146 124 154)(117 145 125 153)(118 160 126 152)(119 159 127 151)(120 158 128 150)
(1 62 23 134 114)(2 63 24 135 115)(3 64 25 136 116)(4 49 26 137 117)(5 50 27 138 118)(6 51 28 139 119)(7 52 29 140 120)(8 53 30 141 121)(9 54 31 142 122)(10 55 32 143 123)(11 56 17 144 124)(12 57 18 129 125)(13 58 19 130 126)(14 59 20 131 127)(15 60 21 132 128)(16 61 22 133 113)(33 111 148 80 85)(34 112 149 65 86)(35 97 150 66 87)(36 98 151 67 88)(37 99 152 68 89)(38 100 153 69 90)(39 101 154 70 91)(40 102 155 71 92)(41 103 156 72 93)(42 104 157 73 94)(43 105 158 74 95)(44 106 159 75 96)(45 107 160 76 81)(46 108 145 77 82)(47 109 146 78 83)(48 110 147 79 84)
(1 156)(2 149)(3 158)(4 151)(5 160)(6 153)(7 146)(8 155)(9 148)(10 157)(11 150)(12 159)(13 152)(14 145)(15 154)(16 147)(17 35)(18 44)(19 37)(20 46)(21 39)(22 48)(23 41)(24 34)(25 43)(26 36)(27 45)(28 38)(29 47)(30 40)(31 33)(32 42)(49 98)(50 107)(51 100)(52 109)(53 102)(54 111)(55 104)(56 97)(57 106)(58 99)(59 108)(60 101)(61 110)(62 103)(63 112)(64 105)(65 115)(66 124)(67 117)(68 126)(69 119)(70 128)(71 121)(72 114)(73 123)(74 116)(75 125)(76 118)(77 127)(78 120)(79 113)(80 122)(81 138)(82 131)(83 140)(84 133)(85 142)(86 135)(87 144)(88 137)(89 130)(90 139)(91 132)(92 141)(93 134)(94 143)(95 136)(96 129)
G:=sub<Sym(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), (1,80,9,72)(2,79,10,71)(3,78,11,70)(4,77,12,69)(5,76,13,68)(6,75,14,67)(7,74,15,66)(8,73,16,65)(17,39,25,47)(18,38,26,46)(19,37,27,45)(20,36,28,44)(21,35,29,43)(22,34,30,42)(23,33,31,41)(24,48,32,40)(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)(97,140,105,132)(98,139,106,131)(99,138,107,130)(100,137,108,129)(101,136,109,144)(102,135,110,143)(103,134,111,142)(104,133,112,141)(113,149,121,157)(114,148,122,156)(115,147,123,155)(116,146,124,154)(117,145,125,153)(118,160,126,152)(119,159,127,151)(120,158,128,150), (1,62,23,134,114)(2,63,24,135,115)(3,64,25,136,116)(4,49,26,137,117)(5,50,27,138,118)(6,51,28,139,119)(7,52,29,140,120)(8,53,30,141,121)(9,54,31,142,122)(10,55,32,143,123)(11,56,17,144,124)(12,57,18,129,125)(13,58,19,130,126)(14,59,20,131,127)(15,60,21,132,128)(16,61,22,133,113)(33,111,148,80,85)(34,112,149,65,86)(35,97,150,66,87)(36,98,151,67,88)(37,99,152,68,89)(38,100,153,69,90)(39,101,154,70,91)(40,102,155,71,92)(41,103,156,72,93)(42,104,157,73,94)(43,105,158,74,95)(44,106,159,75,96)(45,107,160,76,81)(46,108,145,77,82)(47,109,146,78,83)(48,110,147,79,84), (1,156)(2,149)(3,158)(4,151)(5,160)(6,153)(7,146)(8,155)(9,148)(10,157)(11,150)(12,159)(13,152)(14,145)(15,154)(16,147)(17,35)(18,44)(19,37)(20,46)(21,39)(22,48)(23,41)(24,34)(25,43)(26,36)(27,45)(28,38)(29,47)(30,40)(31,33)(32,42)(49,98)(50,107)(51,100)(52,109)(53,102)(54,111)(55,104)(56,97)(57,106)(58,99)(59,108)(60,101)(61,110)(62,103)(63,112)(64,105)(65,115)(66,124)(67,117)(68,126)(69,119)(70,128)(71,121)(72,114)(73,123)(74,116)(75,125)(76,118)(77,127)(78,120)(79,113)(80,122)(81,138)(82,131)(83,140)(84,133)(85,142)(86,135)(87,144)(88,137)(89,130)(90,139)(91,132)(92,141)(93,134)(94,143)(95,136)(96,129)>;
G:=Group( (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), (1,80,9,72)(2,79,10,71)(3,78,11,70)(4,77,12,69)(5,76,13,68)(6,75,14,67)(7,74,15,66)(8,73,16,65)(17,39,25,47)(18,38,26,46)(19,37,27,45)(20,36,28,44)(21,35,29,43)(22,34,30,42)(23,33,31,41)(24,48,32,40)(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)(97,140,105,132)(98,139,106,131)(99,138,107,130)(100,137,108,129)(101,136,109,144)(102,135,110,143)(103,134,111,142)(104,133,112,141)(113,149,121,157)(114,148,122,156)(115,147,123,155)(116,146,124,154)(117,145,125,153)(118,160,126,152)(119,159,127,151)(120,158,128,150), (1,62,23,134,114)(2,63,24,135,115)(3,64,25,136,116)(4,49,26,137,117)(5,50,27,138,118)(6,51,28,139,119)(7,52,29,140,120)(8,53,30,141,121)(9,54,31,142,122)(10,55,32,143,123)(11,56,17,144,124)(12,57,18,129,125)(13,58,19,130,126)(14,59,20,131,127)(15,60,21,132,128)(16,61,22,133,113)(33,111,148,80,85)(34,112,149,65,86)(35,97,150,66,87)(36,98,151,67,88)(37,99,152,68,89)(38,100,153,69,90)(39,101,154,70,91)(40,102,155,71,92)(41,103,156,72,93)(42,104,157,73,94)(43,105,158,74,95)(44,106,159,75,96)(45,107,160,76,81)(46,108,145,77,82)(47,109,146,78,83)(48,110,147,79,84), (1,156)(2,149)(3,158)(4,151)(5,160)(6,153)(7,146)(8,155)(9,148)(10,157)(11,150)(12,159)(13,152)(14,145)(15,154)(16,147)(17,35)(18,44)(19,37)(20,46)(21,39)(22,48)(23,41)(24,34)(25,43)(26,36)(27,45)(28,38)(29,47)(30,40)(31,33)(32,42)(49,98)(50,107)(51,100)(52,109)(53,102)(54,111)(55,104)(56,97)(57,106)(58,99)(59,108)(60,101)(61,110)(62,103)(63,112)(64,105)(65,115)(66,124)(67,117)(68,126)(69,119)(70,128)(71,121)(72,114)(73,123)(74,116)(75,125)(76,118)(77,127)(78,120)(79,113)(80,122)(81,138)(82,131)(83,140)(84,133)(85,142)(86,135)(87,144)(88,137)(89,130)(90,139)(91,132)(92,141)(93,134)(94,143)(95,136)(96,129) );
G=PermutationGroup([[(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)], [(1,80,9,72),(2,79,10,71),(3,78,11,70),(4,77,12,69),(5,76,13,68),(6,75,14,67),(7,74,15,66),(8,73,16,65),(17,39,25,47),(18,38,26,46),(19,37,27,45),(20,36,28,44),(21,35,29,43),(22,34,30,42),(23,33,31,41),(24,48,32,40),(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),(97,140,105,132),(98,139,106,131),(99,138,107,130),(100,137,108,129),(101,136,109,144),(102,135,110,143),(103,134,111,142),(104,133,112,141),(113,149,121,157),(114,148,122,156),(115,147,123,155),(116,146,124,154),(117,145,125,153),(118,160,126,152),(119,159,127,151),(120,158,128,150)], [(1,62,23,134,114),(2,63,24,135,115),(3,64,25,136,116),(4,49,26,137,117),(5,50,27,138,118),(6,51,28,139,119),(7,52,29,140,120),(8,53,30,141,121),(9,54,31,142,122),(10,55,32,143,123),(11,56,17,144,124),(12,57,18,129,125),(13,58,19,130,126),(14,59,20,131,127),(15,60,21,132,128),(16,61,22,133,113),(33,111,148,80,85),(34,112,149,65,86),(35,97,150,66,87),(36,98,151,67,88),(37,99,152,68,89),(38,100,153,69,90),(39,101,154,70,91),(40,102,155,71,92),(41,103,156,72,93),(42,104,157,73,94),(43,105,158,74,95),(44,106,159,75,96),(45,107,160,76,81),(46,108,145,77,82),(47,109,146,78,83),(48,110,147,79,84)], [(1,156),(2,149),(3,158),(4,151),(5,160),(6,153),(7,146),(8,155),(9,148),(10,157),(11,150),(12,159),(13,152),(14,145),(15,154),(16,147),(17,35),(18,44),(19,37),(20,46),(21,39),(22,48),(23,41),(24,34),(25,43),(26,36),(27,45),(28,38),(29,47),(30,40),(31,33),(32,42),(49,98),(50,107),(51,100),(52,109),(53,102),(54,111),(55,104),(56,97),(57,106),(58,99),(59,108),(60,101),(61,110),(62,103),(63,112),(64,105),(65,115),(66,124),(67,117),(68,126),(69,119),(70,128),(71,121),(72,114),(73,123),(74,116),(75,125),(76,118),(77,127),(78,120),(79,113),(80,122),(81,138),(82,131),(83,140),(84,133),(85,142),(86,135),(87,144),(88,137),(89,130),(90,139),(91,132),(92,141),(93,134),(94,143),(95,136),(96,129)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 40 | 2 | 8 | 8 | 10 | 40 | 2 | 2 | 2 | 2 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 16 | 16 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | Q32⋊C2 | D4×D5 | D5×D8 | Q32⋊D5 |
| kernel | Q32⋊D5 | C80⋊C2 | C16⋊D5 | C5⋊SD32 | C5⋊Q32 | C5×Q32 | D5×Q16 | Q8.D10 | C5⋊2C8 | C4×D5 | Q32 | Dic5 | D10 | C16 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 |
Matrix representation of Q32⋊D5 ►in GL4(𝔽241) generated by
| 47 | 173 | 124 | 118 |
| 68 | 194 | 123 | 117 |
| 117 | 123 | 47 | 173 |
| 118 | 124 | 68 | 194 |
| 58 | 234 | 145 | 0 |
| 7 | 183 | 0 | 145 |
| 145 | 0 | 183 | 7 |
| 0 | 145 | 234 | 58 |
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 51 |
| 7 | 183 | 0 | 145 |
| 58 | 234 | 145 | 0 |
| 0 | 96 | 7 | 183 |
| 96 | 0 | 58 | 234 |
G:=sub<GL(4,GF(241))| [47,68,117,118,173,194,123,124,124,123,47,68,118,117,173,194],[58,7,145,0,234,183,0,145,145,0,183,234,0,145,7,58],[0,240,0,0,1,51,0,0,0,0,0,240,0,0,1,51],[7,58,0,96,183,234,96,0,0,145,7,58,145,0,183,234] >;
Q32⋊D5 in GAP, Magma, Sage, TeX
Q_{32}\rtimes D_5 % in TeX
G:=Group("Q32:D5"); // GroupNames label
G:=SmallGroup(320,545);
// by ID
G=gap.SmallGroup(320,545);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=c^5=d^2=1,b^2=a^8,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations