metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.12SD16, C4.Q8⋊6D5, C4⋊C4.36D10, (C2×C8).137D10, C2.22(D5×SD16), C20.27(C4○D4), C4.72(C4○D20), C10.Q16⋊15C2, D10⋊2Q8.5C2, C20.Q8⋊16C2, (C2×Dic5).48D4, C10.38(C2×SD16), C22.214(D4×D5), D10⋊1C8.13C2, C20.44D4⋊31C2, (C2×C20).278C23, (C2×C40).284C22, C4.24(Q8⋊2D5), (C22×D5).119D4, C5⋊3(C23.47D4), C2.23(SD16⋊D5), C10.42(C8.C22), C4⋊Dic5.110C22, (C2×Dic10).86C22, C2.11(D10.13D4), C10.41(C22.D4), (D5×C4⋊C4).6C2, (C5×C4.Q8)⋊15C2, (C2×C4×D5).35C22, (C2×C10).283(C2×D4), (C5×C4⋊C4).71C22, (C2×C5⋊2C8).56C22, (C2×C4).381(C22×D5), SmallGroup(320,489)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.12SD16
G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c3 >
Subgroups: 430 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×9], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C22×C4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C4.Q8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, C40, Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.47D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, C20.Q8, C10.Q16, C20.44D4, D10⋊1C8, C5×C4.Q8, D5×C4⋊C4, D10⋊2Q8, D10.12SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8.C22, C22×D5, C23.47D4, C4○D20, D4×D5, Q8⋊2D5, D10.13D4, D5×SD16, SD16⋊D5, D10.12SD16
►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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 50)(21 145)(22 144)(23 143)(24 142)(25 141)(26 150)(27 149)(28 148)(29 147)(30 146)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 124)(102 123)(103 122)(104 121)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 160)(140 159)
(1 90 50 73 33 66 11 52)(2 81 41 74 34 67 12 53)(3 82 42 75 35 68 13 54)(4 83 43 76 36 69 14 55)(5 84 44 77 37 70 15 56)(6 85 45 78 38 61 16 57)(7 86 46 79 39 62 17 58)(8 87 47 80 40 63 18 59)(9 88 48 71 31 64 19 60)(10 89 49 72 32 65 20 51)(21 118 137 125 141 97 158 101)(22 119 138 126 142 98 159 102)(23 120 139 127 143 99 160 103)(24 111 140 128 144 100 151 104)(25 112 131 129 145 91 152 105)(26 113 132 130 146 92 153 106)(27 114 133 121 147 93 154 107)(28 115 134 122 148 94 155 108)(29 116 135 123 149 95 156 109)(30 117 136 124 150 96 157 110)
(1 97 6 92)(2 98 7 93)(3 99 8 94)(4 100 9 95)(5 91 10 96)(11 101 16 106)(12 102 17 107)(13 103 18 108)(14 104 19 109)(15 105 20 110)(21 85 26 90)(22 86 27 81)(23 87 28 82)(24 88 29 83)(25 89 30 84)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,145)(22,144)(23,143)(24,142)(25,141)(26,150)(27,149)(28,148)(29,147)(30,146)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,113)(92,112)(93,111)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,124)(102,123)(103,122)(104,121)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,160)(140,159), (1,90,50,73,33,66,11,52)(2,81,41,74,34,67,12,53)(3,82,42,75,35,68,13,54)(4,83,43,76,36,69,14,55)(5,84,44,77,37,70,15,56)(6,85,45,78,38,61,16,57)(7,86,46,79,39,62,17,58)(8,87,47,80,40,63,18,59)(9,88,48,71,31,64,19,60)(10,89,49,72,32,65,20,51)(21,118,137,125,141,97,158,101)(22,119,138,126,142,98,159,102)(23,120,139,127,143,99,160,103)(24,111,140,128,144,100,151,104)(25,112,131,129,145,91,152,105)(26,113,132,130,146,92,153,106)(27,114,133,121,147,93,154,107)(28,115,134,122,148,94,155,108)(29,116,135,123,149,95,156,109)(30,117,136,124,150,96,157,110), (1,97,6,92)(2,98,7,93)(3,99,8,94)(4,100,9,95)(5,91,10,96)(11,101,16,106)(12,102,17,107)(13,103,18,108)(14,104,19,109)(15,105,20,110)(21,85,26,90)(22,86,27,81)(23,87,28,82)(24,88,29,83)(25,89,30,84)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155)>;
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,145)(22,144)(23,143)(24,142)(25,141)(26,150)(27,149)(28,148)(29,147)(30,146)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,80)(60,79)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,113)(92,112)(93,111)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,124)(102,123)(103,122)(104,121)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,160)(140,159), (1,90,50,73,33,66,11,52)(2,81,41,74,34,67,12,53)(3,82,42,75,35,68,13,54)(4,83,43,76,36,69,14,55)(5,84,44,77,37,70,15,56)(6,85,45,78,38,61,16,57)(7,86,46,79,39,62,17,58)(8,87,47,80,40,63,18,59)(9,88,48,71,31,64,19,60)(10,89,49,72,32,65,20,51)(21,118,137,125,141,97,158,101)(22,119,138,126,142,98,159,102)(23,120,139,127,143,99,160,103)(24,111,140,128,144,100,151,104)(25,112,131,129,145,91,152,105)(26,113,132,130,146,92,153,106)(27,114,133,121,147,93,154,107)(28,115,134,122,148,94,155,108)(29,116,135,123,149,95,156,109)(30,117,136,124,150,96,157,110), (1,97,6,92)(2,98,7,93)(3,99,8,94)(4,100,9,95)(5,91,10,96)(11,101,16,106)(12,102,17,107)(13,103,18,108)(14,104,19,109)(15,105,20,110)(21,85,26,90)(22,86,27,81)(23,87,28,82)(24,88,29,83)(25,89,30,84)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155) );
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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,50),(21,145),(22,144),(23,143),(24,142),(25,141),(26,150),(27,149),(28,148),(29,147),(30,146),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,80),(60,79),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,90),(91,113),(92,112),(93,111),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,124),(102,123),(103,122),(104,121),(105,130),(106,129),(107,128),(108,127),(109,126),(110,125),(131,158),(132,157),(133,156),(134,155),(135,154),(136,153),(137,152),(138,151),(139,160),(140,159)], [(1,90,50,73,33,66,11,52),(2,81,41,74,34,67,12,53),(3,82,42,75,35,68,13,54),(4,83,43,76,36,69,14,55),(5,84,44,77,37,70,15,56),(6,85,45,78,38,61,16,57),(7,86,46,79,39,62,17,58),(8,87,47,80,40,63,18,59),(9,88,48,71,31,64,19,60),(10,89,49,72,32,65,20,51),(21,118,137,125,141,97,158,101),(22,119,138,126,142,98,159,102),(23,120,139,127,143,99,160,103),(24,111,140,128,144,100,151,104),(25,112,131,129,145,91,152,105),(26,113,132,130,146,92,153,106),(27,114,133,121,147,93,154,107),(28,115,134,122,148,94,155,108),(29,116,135,123,149,95,156,109),(30,117,136,124,150,96,157,110)], [(1,97,6,92),(2,98,7,93),(3,99,8,94),(4,100,9,95),(5,91,10,96),(11,101,16,106),(12,102,17,107),(13,103,18,108),(14,104,19,109),(15,105,20,110),(21,85,26,90),(22,86,27,81),(23,87,28,82),(24,88,29,83),(25,89,30,84),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | C4○D20 | C8.C22 | Q8⋊2D5 | D4×D5 | D5×SD16 | SD16⋊D5 |
| kernel | D10.12SD16 | C20.Q8 | C10.Q16 | C20.44D4 | D10⋊1C8 | C5×C4.Q8 | D5×C4⋊C4 | D10⋊2Q8 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | D10 | C4⋊C4 | C2×C8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10.12SD16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 34 |
| 0 | 0 | 7 | 34 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 7 | 40 |
| 15 | 26 | 0 | 0 |
| 15 | 15 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 17 | 32 | 0 | 0 |
| 32 | 24 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,7,0,0,34,34],[40,0,0,0,0,40,0,0,0,0,1,7,0,0,0,40],[15,15,0,0,26,15,0,0,0,0,17,40,0,0,1,24],[17,32,0,0,32,24,0,0,0,0,9,0,0,0,0,9] >;
D10.12SD16 in GAP, Magma, Sage, TeX
D_{10}._{12}{\rm SD}_{16} % in TeX
G:=Group("D10.12SD16"); // GroupNames label
G:=SmallGroup(320,489);
// by ID
G=gap.SmallGroup(320,489);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,100,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations