metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD32⋊2D5, D8.4D10, C16.2D10, Dic40⋊6C2, D10.16D8, C80.9C22, Q16.1D10, C40.18C23, Dic5.18D8, Dic20.3C22, C4.6(D4×D5), D8.D5⋊4C2, (D5×Q16)⋊4C2, C5⋊Q32⋊1C2, C80⋊C2⋊2C2, (C4×D5).9D4, C2.21(D5×D8), C5⋊2C8.4D4, (C5×SD32)⋊2C2, C10.37(C2×D8), C20.12(C2×D4), C5⋊2(Q32⋊C2), D8⋊3D5.1C2, (C8×D5).5C22, (C5×D8).4C22, C8.24(C22×D5), C5⋊2C16.1C22, (C5×Q16).2C22, SmallGroup(320,542)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD32⋊D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a7, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 374 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, M5(2), SD32, SD32, Q32, C2×Q16, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, Q32⋊C2, C5⋊2C16, C80, C8×D5, Dic20, D4.D5, C5⋊Q16, C5×D8, C5×Q16, D4⋊2D5, Q8×D5, C80⋊C2, Dic40, D8.D5, C5⋊Q32, C5×SD32, D8⋊3D5, D5×Q16, SD32⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, Q32⋊C2, D4×D5, D5×D8, SD32⋊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)
(1 139)(2 130)(3 137)(4 144)(5 135)(6 142)(7 133)(8 140)(9 131)(10 138)(11 129)(12 136)(13 143)(14 134)(15 141)(16 132)(17 89)(18 96)(19 87)(20 94)(21 85)(22 92)(23 83)(24 90)(25 81)(26 88)(27 95)(28 86)(29 93)(30 84)(31 91)(32 82)(33 147)(34 154)(35 145)(36 152)(37 159)(38 150)(39 157)(40 148)(41 155)(42 146)(43 153)(44 160)(45 151)(46 158)(47 149)(48 156)(49 111)(50 102)(51 109)(52 100)(53 107)(54 98)(55 105)(56 112)(57 103)(58 110)(59 101)(60 108)(61 99)(62 106)(63 97)(64 104)(65 115)(66 122)(67 113)(68 120)(69 127)(70 118)(71 125)(72 116)(73 123)(74 114)(75 121)(76 128)(77 119)(78 126)(79 117)(80 124)
(1 61 19 159 79)(2 62 20 160 80)(3 63 21 145 65)(4 64 22 146 66)(5 49 23 147 67)(6 50 24 148 68)(7 51 25 149 69)(8 52 26 150 70)(9 53 27 151 71)(10 54 28 152 72)(11 55 29 153 73)(12 56 30 154 74)(13 57 31 155 75)(14 58 32 156 76)(15 59 17 157 77)(16 60 18 158 78)(33 113 135 111 83)(34 114 136 112 84)(35 115 137 97 85)(36 116 138 98 86)(37 117 139 99 87)(38 118 140 100 88)(39 119 141 101 89)(40 120 142 102 90)(41 121 143 103 91)(42 122 144 104 92)(43 123 129 105 93)(44 124 130 106 94)(45 125 131 107 95)(46 126 132 108 96)(47 127 133 109 81)(48 128 134 110 82)
(1 79)(2 72)(3 65)(4 74)(5 67)(6 76)(7 69)(8 78)(9 71)(10 80)(11 73)(12 66)(13 75)(14 68)(15 77)(16 70)(18 26)(20 28)(22 30)(24 32)(33 103)(34 112)(35 105)(36 98)(37 107)(38 100)(39 109)(40 102)(41 111)(42 104)(43 97)(44 106)(45 99)(46 108)(47 101)(48 110)(49 147)(50 156)(51 149)(52 158)(53 151)(54 160)(55 153)(56 146)(57 155)(58 148)(59 157)(60 150)(61 159)(62 152)(63 145)(64 154)(81 89)(83 91)(85 93)(87 95)(113 143)(114 136)(115 129)(116 138)(117 131)(118 140)(119 133)(120 142)(121 135)(122 144)(123 137)(124 130)(125 139)(126 132)(127 141)(128 134)
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), (1,139)(2,130)(3,137)(4,144)(5,135)(6,142)(7,133)(8,140)(9,131)(10,138)(11,129)(12,136)(13,143)(14,134)(15,141)(16,132)(17,89)(18,96)(19,87)(20,94)(21,85)(22,92)(23,83)(24,90)(25,81)(26,88)(27,95)(28,86)(29,93)(30,84)(31,91)(32,82)(33,147)(34,154)(35,145)(36,152)(37,159)(38,150)(39,157)(40,148)(41,155)(42,146)(43,153)(44,160)(45,151)(46,158)(47,149)(48,156)(49,111)(50,102)(51,109)(52,100)(53,107)(54,98)(55,105)(56,112)(57,103)(58,110)(59,101)(60,108)(61,99)(62,106)(63,97)(64,104)(65,115)(66,122)(67,113)(68,120)(69,127)(70,118)(71,125)(72,116)(73,123)(74,114)(75,121)(76,128)(77,119)(78,126)(79,117)(80,124), (1,61,19,159,79)(2,62,20,160,80)(3,63,21,145,65)(4,64,22,146,66)(5,49,23,147,67)(6,50,24,148,68)(7,51,25,149,69)(8,52,26,150,70)(9,53,27,151,71)(10,54,28,152,72)(11,55,29,153,73)(12,56,30,154,74)(13,57,31,155,75)(14,58,32,156,76)(15,59,17,157,77)(16,60,18,158,78)(33,113,135,111,83)(34,114,136,112,84)(35,115,137,97,85)(36,116,138,98,86)(37,117,139,99,87)(38,118,140,100,88)(39,119,141,101,89)(40,120,142,102,90)(41,121,143,103,91)(42,122,144,104,92)(43,123,129,105,93)(44,124,130,106,94)(45,125,131,107,95)(46,126,132,108,96)(47,127,133,109,81)(48,128,134,110,82), (1,79)(2,72)(3,65)(4,74)(5,67)(6,76)(7,69)(8,78)(9,71)(10,80)(11,73)(12,66)(13,75)(14,68)(15,77)(16,70)(18,26)(20,28)(22,30)(24,32)(33,103)(34,112)(35,105)(36,98)(37,107)(38,100)(39,109)(40,102)(41,111)(42,104)(43,97)(44,106)(45,99)(46,108)(47,101)(48,110)(49,147)(50,156)(51,149)(52,158)(53,151)(54,160)(55,153)(56,146)(57,155)(58,148)(59,157)(60,150)(61,159)(62,152)(63,145)(64,154)(81,89)(83,91)(85,93)(87,95)(113,143)(114,136)(115,129)(116,138)(117,131)(118,140)(119,133)(120,142)(121,135)(122,144)(123,137)(124,130)(125,139)(126,132)(127,141)(128,134)>;
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), (1,139)(2,130)(3,137)(4,144)(5,135)(6,142)(7,133)(8,140)(9,131)(10,138)(11,129)(12,136)(13,143)(14,134)(15,141)(16,132)(17,89)(18,96)(19,87)(20,94)(21,85)(22,92)(23,83)(24,90)(25,81)(26,88)(27,95)(28,86)(29,93)(30,84)(31,91)(32,82)(33,147)(34,154)(35,145)(36,152)(37,159)(38,150)(39,157)(40,148)(41,155)(42,146)(43,153)(44,160)(45,151)(46,158)(47,149)(48,156)(49,111)(50,102)(51,109)(52,100)(53,107)(54,98)(55,105)(56,112)(57,103)(58,110)(59,101)(60,108)(61,99)(62,106)(63,97)(64,104)(65,115)(66,122)(67,113)(68,120)(69,127)(70,118)(71,125)(72,116)(73,123)(74,114)(75,121)(76,128)(77,119)(78,126)(79,117)(80,124), (1,61,19,159,79)(2,62,20,160,80)(3,63,21,145,65)(4,64,22,146,66)(5,49,23,147,67)(6,50,24,148,68)(7,51,25,149,69)(8,52,26,150,70)(9,53,27,151,71)(10,54,28,152,72)(11,55,29,153,73)(12,56,30,154,74)(13,57,31,155,75)(14,58,32,156,76)(15,59,17,157,77)(16,60,18,158,78)(33,113,135,111,83)(34,114,136,112,84)(35,115,137,97,85)(36,116,138,98,86)(37,117,139,99,87)(38,118,140,100,88)(39,119,141,101,89)(40,120,142,102,90)(41,121,143,103,91)(42,122,144,104,92)(43,123,129,105,93)(44,124,130,106,94)(45,125,131,107,95)(46,126,132,108,96)(47,127,133,109,81)(48,128,134,110,82), (1,79)(2,72)(3,65)(4,74)(5,67)(6,76)(7,69)(8,78)(9,71)(10,80)(11,73)(12,66)(13,75)(14,68)(15,77)(16,70)(18,26)(20,28)(22,30)(24,32)(33,103)(34,112)(35,105)(36,98)(37,107)(38,100)(39,109)(40,102)(41,111)(42,104)(43,97)(44,106)(45,99)(46,108)(47,101)(48,110)(49,147)(50,156)(51,149)(52,158)(53,151)(54,160)(55,153)(56,146)(57,155)(58,148)(59,157)(60,150)(61,159)(62,152)(63,145)(64,154)(81,89)(83,91)(85,93)(87,95)(113,143)(114,136)(115,129)(116,138)(117,131)(118,140)(119,133)(120,142)(121,135)(122,144)(123,137)(124,130)(125,139)(126,132)(127,141)(128,134) );
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)], [(1,139),(2,130),(3,137),(4,144),(5,135),(6,142),(7,133),(8,140),(9,131),(10,138),(11,129),(12,136),(13,143),(14,134),(15,141),(16,132),(17,89),(18,96),(19,87),(20,94),(21,85),(22,92),(23,83),(24,90),(25,81),(26,88),(27,95),(28,86),(29,93),(30,84),(31,91),(32,82),(33,147),(34,154),(35,145),(36,152),(37,159),(38,150),(39,157),(40,148),(41,155),(42,146),(43,153),(44,160),(45,151),(46,158),(47,149),(48,156),(49,111),(50,102),(51,109),(52,100),(53,107),(54,98),(55,105),(56,112),(57,103),(58,110),(59,101),(60,108),(61,99),(62,106),(63,97),(64,104),(65,115),(66,122),(67,113),(68,120),(69,127),(70,118),(71,125),(72,116),(73,123),(74,114),(75,121),(76,128),(77,119),(78,126),(79,117),(80,124)], [(1,61,19,159,79),(2,62,20,160,80),(3,63,21,145,65),(4,64,22,146,66),(5,49,23,147,67),(6,50,24,148,68),(7,51,25,149,69),(8,52,26,150,70),(9,53,27,151,71),(10,54,28,152,72),(11,55,29,153,73),(12,56,30,154,74),(13,57,31,155,75),(14,58,32,156,76),(15,59,17,157,77),(16,60,18,158,78),(33,113,135,111,83),(34,114,136,112,84),(35,115,137,97,85),(36,116,138,98,86),(37,117,139,99,87),(38,118,140,100,88),(39,119,141,101,89),(40,120,142,102,90),(41,121,143,103,91),(42,122,144,104,92),(43,123,129,105,93),(44,124,130,106,94),(45,125,131,107,95),(46,126,132,108,96),(47,127,133,109,81),(48,128,134,110,82)], [(1,79),(2,72),(3,65),(4,74),(5,67),(6,76),(7,69),(8,78),(9,71),(10,80),(11,73),(12,66),(13,75),(14,68),(15,77),(16,70),(18,26),(20,28),(22,30),(24,32),(33,103),(34,112),(35,105),(36,98),(37,107),(38,100),(39,109),(40,102),(41,111),(42,104),(43,97),(44,106),(45,99),(46,108),(47,101),(48,110),(49,147),(50,156),(51,149),(52,158),(53,151),(54,160),(55,153),(56,146),(57,155),(58,148),(59,157),(60,150),(61,159),(62,152),(63,145),(64,154),(81,89),(83,91),(85,93),(87,95),(113,143),(114,136),(115,129),(116,138),(117,131),(118,140),(119,133),(120,142),(121,135),(122,144),(123,137),(124,130),(125,139),(126,132),(127,141),(128,134)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 10 | 2 | 8 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 20 | 2 | 2 | 16 | 16 | 4 | 4 | 20 | 20 | 4 | 4 | 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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | Q32⋊C2 | D4×D5 | D5×D8 | SD32⋊D5 |
| kernel | SD32⋊D5 | C80⋊C2 | Dic40 | D8.D5 | C5⋊Q32 | C5×SD32 | D8⋊3D5 | D5×Q16 | C5⋊2C8 | C4×D5 | SD32 | Dic5 | D10 | C16 | D8 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of SD32⋊D5 ►in GL4(𝔽241) generated by
| 32 | 110 | 168 | 133 |
| 131 | 209 | 168 | 168 |
| 103 | 49 | 228 | 120 |
| 138 | 103 | 121 | 13 |
| 208 | 233 | 80 | 224 |
| 8 | 33 | 80 | 80 |
| 141 | 39 | 208 | 8 |
| 100 | 141 | 233 | 33 |
| 189 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 240 | 52 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [32,131,103,138,110,209,49,103,168,168,228,121,133,168,120,13],[208,8,141,100,233,33,39,141,80,80,208,233,224,80,8,33],[189,240,0,0,1,0,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,52,1,0,0,0,0,0,240,0,0,240,0] >;
SD32⋊D5 in GAP, Magma, Sage, TeX
{\rm SD}_{32}\rtimes D_5 % in TeX
G:=Group("SD32:D5"); // GroupNames label
G:=SmallGroup(320,542);
// by ID
G=gap.SmallGroup(320,542);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,135,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^7,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