direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C2×C10, C40⋊14C23, C20.79C24, C8⋊3(C22×C10), C4.19(D4×C10), (C22×C8)⋊10C10, (C22×C40)⋊24C2, (C2×C40)⋊52C22, (C2×C20).433D4, C20.326(C2×D4), C4.2(C23×C10), (C5×Q8)⋊11C23, Q8⋊1(C22×C10), (C22×Q8)⋊8C10, C23.61(C5×D4), (Q8×C10)⋊53C22, D4.1(C22×C10), (C5×D4).34C23, C22.66(D4×C10), (C2×C20).972C23, (C22×D4).12C10, (C22×C10).222D4, C10.200(C22×D4), (D4×C10).327C22, (C22×C20).602C22, (Q8×C2×C10)⋊20C2, C2.24(D4×C2×C10), (C2×C8)⋊14(C2×C10), (D4×C2×C10).25C2, (C2×C4).89(C5×D4), (C2×Q8)⋊13(C2×C10), (C2×D4).73(C2×C10), (C2×C10).687(C2×D4), (C22×C4).129(C2×C10), (C2×C4).142(C22×C10), SmallGroup(320,1572)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C2×C10
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 498 in 298 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C20, C20, C20, C2×C10, C2×C10, C22×C8, C2×SD16, C22×D4, C22×Q8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C22×SD16, C2×C40, C5×SD16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, Q8×C10, C23×C10, C22×C40, C10×SD16, D4×C2×C10, Q8×C2×C10, SD16×C2×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C24, C2×C10, C2×SD16, C22×D4, C5×D4, C22×C10, C22×SD16, C5×SD16, D4×C10, C23×C10, C10×SD16, D4×C2×C10, SD16×C2×C10
►On 160 points
Generators in S160
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 146)(12 147)(13 148)(14 149)(15 150)(16 141)(17 142)(18 143)(19 144)(20 145)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 111)(31 140)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 101)(60 102)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 151)(130 152)
(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)
(1 153 41 11 59 27 69 40)(2 154 42 12 60 28 70 31)(3 155 43 13 51 29 61 32)(4 156 44 14 52 30 62 33)(5 157 45 15 53 21 63 34)(6 158 46 16 54 22 64 35)(7 159 47 17 55 23 65 36)(8 160 48 18 56 24 66 37)(9 151 49 19 57 25 67 38)(10 152 50 20 58 26 68 39)(71 144 109 116 84 137 92 129)(72 145 110 117 85 138 93 130)(73 146 101 118 86 139 94 121)(74 147 102 119 87 140 95 122)(75 148 103 120 88 131 96 123)(76 149 104 111 89 132 97 124)(77 150 105 112 90 133 98 125)(78 141 106 113 81 134 99 126)(79 142 107 114 82 135 100 127)(80 143 108 115 83 136 91 128)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 61)(49 62)(50 63)(51 56)(52 57)(53 58)(54 59)(55 60)(71 89)(72 90)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 137)(112 138)(113 139)(114 140)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)
G:=sub<Sym(160)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,146)(12,147)(13,148)(14,149)(15,150)(16,141)(17,142)(18,143)(19,144)(20,145)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,140)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,71)(50,72)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,151)(130,152), (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), (1,153,41,11,59,27,69,40)(2,154,42,12,60,28,70,31)(3,155,43,13,51,29,61,32)(4,156,44,14,52,30,62,33)(5,157,45,15,53,21,63,34)(6,158,46,16,54,22,64,35)(7,159,47,17,55,23,65,36)(8,160,48,18,56,24,66,37)(9,151,49,19,57,25,67,38)(10,152,50,20,58,26,68,39)(71,144,109,116,84,137,92,129)(72,145,110,117,85,138,93,130)(73,146,101,118,86,139,94,121)(74,147,102,119,87,140,95,122)(75,148,103,120,88,131,96,123)(76,149,104,111,89,132,97,124)(77,150,105,112,90,133,98,125)(78,141,106,113,81,134,99,126)(79,142,107,114,82,135,100,127)(80,143,108,115,83,136,91,128), (1,6)(2,7)(3,8)(4,9)(5,10)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,61)(49,62)(50,63)(51,56)(52,57)(53,58)(54,59)(55,60)(71,89)(72,90)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,137)(112,138)(113,139)(114,140)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)>;
G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,146)(12,147)(13,148)(14,149)(15,150)(16,141)(17,142)(18,143)(19,144)(20,145)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,140)(32,131)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,71)(50,72)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,151)(130,152), (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), (1,153,41,11,59,27,69,40)(2,154,42,12,60,28,70,31)(3,155,43,13,51,29,61,32)(4,156,44,14,52,30,62,33)(5,157,45,15,53,21,63,34)(6,158,46,16,54,22,64,35)(7,159,47,17,55,23,65,36)(8,160,48,18,56,24,66,37)(9,151,49,19,57,25,67,38)(10,152,50,20,58,26,68,39)(71,144,109,116,84,137,92,129)(72,145,110,117,85,138,93,130)(73,146,101,118,86,139,94,121)(74,147,102,119,87,140,95,122)(75,148,103,120,88,131,96,123)(76,149,104,111,89,132,97,124)(77,150,105,112,90,133,98,125)(78,141,106,113,81,134,99,126)(79,142,107,114,82,135,100,127)(80,143,108,115,83,136,91,128), (1,6)(2,7)(3,8)(4,9)(5,10)(11,158)(12,159)(13,160)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,61)(49,62)(50,63)(51,56)(52,57)(53,58)(54,59)(55,60)(71,89)(72,90)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,137)(112,138)(113,139)(114,140)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150) );
G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,91),(9,92),(10,93),(11,146),(12,147),(13,148),(14,149),(15,150),(16,141),(17,142),(18,143),(19,144),(20,145),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,111),(31,140),(32,131),(33,132),(34,133),(35,134),(36,135),(37,136),(38,137),(39,138),(40,139),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,71),(50,72),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,101),(60,102),(61,88),(62,89),(63,90),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,151),(130,152)], [(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)], [(1,153,41,11,59,27,69,40),(2,154,42,12,60,28,70,31),(3,155,43,13,51,29,61,32),(4,156,44,14,52,30,62,33),(5,157,45,15,53,21,63,34),(6,158,46,16,54,22,64,35),(7,159,47,17,55,23,65,36),(8,160,48,18,56,24,66,37),(9,151,49,19,57,25,67,38),(10,152,50,20,58,26,68,39),(71,144,109,116,84,137,92,129),(72,145,110,117,85,138,93,130),(73,146,101,118,86,139,94,121),(74,147,102,119,87,140,95,122),(75,148,103,120,88,131,96,123),(76,149,104,111,89,132,97,124),(77,150,105,112,90,133,98,125),(78,141,106,113,81,134,99,126),(79,142,107,114,82,135,100,127),(80,143,108,115,83,136,91,128)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,158),(12,159),(13,160),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,157),(21,39),(22,40),(23,31),(24,32),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,61),(49,62),(50,63),(51,56),(52,57),(53,58),(54,59),(55,60),(71,89),(72,90),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,137),(112,138),(113,139),(114,140),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C5×D4 | C5×D4 | C5×SD16 |
| kernel | SD16×C2×C10 | C22×C40 | C10×SD16 | D4×C2×C10 | Q8×C2×C10 | C22×SD16 | C22×C8 | C2×SD16 | C22×D4 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 4 | 4 | 48 | 4 | 4 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of SD16×C2×C10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 15 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
SD16×C2×C10 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_2\times C_{10} % in TeX
G:=Group("SD16xC2xC10"); // GroupNames label
G:=SmallGroup(320,1572);
// by ID
G=gap.SmallGroup(320,1572);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations