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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C4.Q8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.47D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, 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, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, 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 43)(12 42)(13 41)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 134)(22 133)(23 132)(24 131)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 90)(60 89)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)(91 123)(92 122)(93 121)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 112)(102 111)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 160)(149 159)(150 158)
(1 90 50 73 33 55 15 61)(2 81 41 74 34 56 16 62)(3 82 42 75 35 57 17 63)(4 83 43 76 36 58 18 64)(5 84 44 77 37 59 19 65)(6 85 45 78 38 60 20 66)(7 86 46 79 39 51 11 67)(8 87 47 80 40 52 12 68)(9 88 48 71 31 53 13 69)(10 89 49 72 32 54 14 70)(21 118 146 125 140 106 158 100)(22 119 147 126 131 107 159 91)(23 120 148 127 132 108 160 92)(24 111 149 128 133 109 151 93)(25 112 150 129 134 110 152 94)(26 113 141 130 135 101 153 95)(27 114 142 121 136 102 154 96)(28 115 143 122 137 103 155 97)(29 116 144 123 138 104 156 98)(30 117 145 124 139 105 157 99)
(1 106 6 101)(2 107 7 102)(3 108 8 103)(4 109 9 104)(5 110 10 105)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(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)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,134)(22,133)(23,132)(24,131)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,90)(60,89)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78)(91,123)(92,122)(93,121)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,160)(149,159)(150,158), (1,90,50,73,33,55,15,61)(2,81,41,74,34,56,16,62)(3,82,42,75,35,57,17,63)(4,83,43,76,36,58,18,64)(5,84,44,77,37,59,19,65)(6,85,45,78,38,60,20,66)(7,86,46,79,39,51,11,67)(8,87,47,80,40,52,12,68)(9,88,48,71,31,53,13,69)(10,89,49,72,32,54,14,70)(21,118,146,125,140,106,158,100)(22,119,147,126,131,107,159,91)(23,120,148,127,132,108,160,92)(24,111,149,128,133,109,151,93)(25,112,150,129,134,110,152,94)(26,113,141,130,135,101,153,95)(27,114,142,121,136,102,154,96)(28,115,143,122,137,103,155,97)(29,116,144,123,138,104,156,98)(30,117,145,124,139,105,157,99), (1,106,6,101)(2,107,7,102)(3,108,8,103)(4,109,9,104)(5,110,10,105)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(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)(11,43)(12,42)(13,41)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,134)(22,133)(23,132)(24,131)(25,140)(26,139)(27,138)(28,137)(29,136)(30,135)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,90)(60,89)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78)(91,123)(92,122)(93,121)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,160)(149,159)(150,158), (1,90,50,73,33,55,15,61)(2,81,41,74,34,56,16,62)(3,82,42,75,35,57,17,63)(4,83,43,76,36,58,18,64)(5,84,44,77,37,59,19,65)(6,85,45,78,38,60,20,66)(7,86,46,79,39,51,11,67)(8,87,47,80,40,52,12,68)(9,88,48,71,31,53,13,69)(10,89,49,72,32,54,14,70)(21,118,146,125,140,106,158,100)(22,119,147,126,131,107,159,91)(23,120,148,127,132,108,160,92)(24,111,149,128,133,109,151,93)(25,112,150,129,134,110,152,94)(26,113,141,130,135,101,153,95)(27,114,142,121,136,102,154,96)(28,115,143,122,137,103,155,97)(29,116,144,123,138,104,156,98)(30,117,145,124,139,105,157,99), (1,106,6,101)(2,107,7,102)(3,108,8,103)(4,109,9,104)(5,110,10,105)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(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),(11,43),(12,42),(13,41),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,134),(22,133),(23,132),(24,131),(25,140),(26,139),(27,138),(28,137),(29,136),(30,135),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,90),(60,89),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,80),(69,79),(70,78),(91,123),(92,122),(93,121),(94,130),(95,129),(96,128),(97,127),(98,126),(99,125),(100,124),(101,112),(102,111),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,160),(149,159),(150,158)], [(1,90,50,73,33,55,15,61),(2,81,41,74,34,56,16,62),(3,82,42,75,35,57,17,63),(4,83,43,76,36,58,18,64),(5,84,44,77,37,59,19,65),(6,85,45,78,38,60,20,66),(7,86,46,79,39,51,11,67),(8,87,47,80,40,52,12,68),(9,88,48,71,31,53,13,69),(10,89,49,72,32,54,14,70),(21,118,146,125,140,106,158,100),(22,119,147,126,131,107,159,91),(23,120,148,127,132,108,160,92),(24,111,149,128,133,109,151,93),(25,112,150,129,134,110,152,94),(26,113,141,130,135,101,153,95),(27,114,142,121,136,102,154,96),(28,115,143,122,137,103,155,97),(29,116,144,123,138,104,156,98),(30,117,145,124,139,105,157,99)], [(1,106,6,101),(2,107,7,102),(3,108,8,103),(4,109,9,104),(5,110,10,105),(11,96,16,91),(12,97,17,92),(13,98,18,93),(14,99,19,94),(15,100,20,95),(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